Why do Sotapannas need faith?

As others have pointed out, faith is regarding faith in others. The majority of the Buddhist thought is actually quite logical, 4 noble truths, 8fold path, dependant origination, 3 poisons and 3 marks of existence can be logically thought of to their conclusions. Originally one would need faith (in the Buddha) to even try to think about these concepts thoroughly.

While a sotapanna has experienced nibbana and knows the first 3 noble truths, he would still need faith in the Buddha to practice the 4th noble truth as it is a path that one has to follow to gain it's fruit and the sotapanna has clearly not walked the entire path. As well as a sotapanna would need or have faith that the Buddha has walked that path and is indeed and arhant from walking that path, as one cannot know teh mind of other beings only oneself.

Knowing that exercise is good is one thing. But exercise is a lot of work. It takes faith.

It takes faith to change comfortable habits. Seeing a teacher abide in peace and joy with body, speech and mind, one might visit them and ask questions about the teaching. Troubled by lingering desires, it takes faith to be satisfied with the spiritual life.

MN68:9.3: Rather, there are gentlemen of faith who are full of sublime joy and gladness.
MN68:5.2: It’s appropriate for gentlemen like yourselves, who have gone forth in faith from the lay life to homelessness, to be satisfied with the spiritual life.

The spiritual life gives up lay comforts. That giving up of relishing is not easy. The spiritual life is supported by rules that restrain and guide. Practicing those rules is not comfortable and requires faith.

MN68:12.7: Recollecting that monk’s faith, ethics, learning, generosity, and wisdom, he applies his mind to that end.
MN68:13.7: Recollecting that monk’s faith, ethics, learning, generosity, and wisdom, he applies his mind to that end.
MN68:20.7: Recollecting that layman’s faith, ethics, learning, generosity, and wisdom, he applies his mind to that end.
MN68:21.7: Recollecting that laywoman’s faith, ethics, learning, generosity, and wisdom, she applies her mind to that end.

Seeing the faith, ethics, learning, generosity and wisdom in others inspires our own faith to practice restraint. Faith encourages us to try harder.

AN9.4:4.6: Nandaka, a mendicant is faithful but not ethical.
AN9.4:4.7: So they’re incomplete in that respect,
AN9.4:4.8: and should fulfill it, thinking,
AN9.4:4.9: ‘How can I become faithful and ethical?’

Having faith in truth is one thing. Faith inspires us to practice that truth.

Knowing that exercise is good is one thing. But exercise is a lot of work. It takes faith.

Having entered the stream, it takes faith to continue to the other shore.

MN34:9.1: Just like the calves and weak cattle who crossed the Ganges to safety
MN34:9.2: are the mendicants who, with the ending of three fetters are stream-enterers, not liable to be reborn in the underworld, bound for awakening. They too, having breasted Māra’s stream, will safely cross over to the far shore.

According to AN 10.92, three of the four factors of stream entry include "experiential confidence" (translated by Ven. Sujato) or "verified confidence" (translated by Ven. Thanissaro) in the Buddha, the Dhamma and the Sangha.

Experiential confidence or verified confidence is different from faith. It means that one has confidence, based on experiential verification.

Why is faith important?

It's answered in this phrase from MN95 - “Faith is most helpful for visiting, Bhāradvāja. If faith in a teacher does not arise, one will not visit him; but because faith in a teacher arises, one visits him. That is why faith is most helpful for visiting.”

The rest of the quoted sutta shows how faith leads to discovery of truth.

“In that way, Master Gotama, there is the discovery of truth; in that way one discovers truth; in that way we recognise the discovery of truth. But in what way, Master Gotama, is there the final arrival at truth? In what way does one finally arrive at truth? We ask Master Gotama about the final arrival at truth.” “The final arrival at truth, Bhāradvāja, lies in the repetition, development, and cultivation of those same things. In this way, Bhāradvāja, there is the final arrival at truth; in this way one finally arrives at truth; in this way we describe the final arrival at truth.”

“In that way, Master Gotama, there is the final arrival at truth; in that way one finally arrives at truth; in that way we recognise the final arrival at truth. But what, Master Gotama, is most helpful for the final arrival at truth? We ask Master Gotama about the thing most helpful for the final arrival at truth.” “Striving is most helpful for the final arrival at truth, Bhāradvāja. If one does not strive, one will not finally arrive at truth; but because one strives, one does finally arrive at truth. That is why striving is most helpful for the final arrival at truth.”

“But what, Master Gotama, is most helpful for striving? We ask Master Gotama about the thing most helpful for striving.”

“Scrutiny is most helpful for striving, Bhāradvāja. If one does not scrutinise, one will not strive; but because one scrutinises, one strives. That is why scrutiny is most helpful for striving.”

“But what, Master Gotama, is most helpful for scrutiny? We ask Master Gotama about the thing most helpful for scrutiny.”

“Application of the will is most helpful for scrutiny, Bhāradvāja. If one does not apply one’s will, one will not scrutinise; but because one applies one’s will, one scrutinises. That is why application of the will is most helpful for scrutiny.”

“But what, Master Gotama, is most helpful for application of the will? We ask Master Gotama about the thing most helpful for application of the will.”

“Zeal is most helpful for application of the will, Bhāradvāja. If one does not arouse zeal, one will not apply one’s will; but because one arouses zeal, one applies one’s will. That is why zeal is most helpful for application of the will.”

“But what, Master Gotama, is most helpful for zeal? We ask Master Gotama about the thing most helpful for zeal.”

“A reflective acceptance of the teachings is most helpful for zeal, Bhāradvāja. If one does not gain a reflective acceptance of the teachings, zeal will not spring up; but because one gains a reflective acceptance of the teachings, zeal springs up. That is why a reflective acceptance of the teachings is most helpful for zeal.”

“But what, Master Gotama, is most helpful for a reflective acceptance of the teachings? We ask Master Gotama about the thing most helpful for a reflective acceptance of the teachings.”

“Examination of the meaning is most helpful for a reflective acceptance of the teachings, Bhāradvāja. If one does not examine their meaning, one will not gain a reflective acceptance of the teachings; but because one examines their meaning, one gains a reflective acceptance of the teachings. That is why examination of the meaning is most helpful for a reflective acceptance of the teachings.”

“But what, Master Gotama, is most helpful for examination of the meaning? We ask Master Gotama about the thing most helpful for examination of meaning.”

“Memorising the teachings is most helpful for examining the meaning, Bhāradvāja. If one does not memorise a teaching, one will not examine its meaning; but because one memorises a teaching, one examines its meaning.”

“But what, Master Gotama, is most helpful for memorising the teachings? We ask Master Gotama about the thing most helpful for memorising the teachings.”

“Hearing the Dhamma is most helpful for memorising the teachings, Bhāradvāja. If one does not hear the Dhamma, one will not memorise the teachings; but because one hears the Dhamma, one memorises the teachings. That is why hearing the Dhamma is most helpful for memorising the teachings.”

“But what, Master Gotama, is most helpful for hearing the Dhamma? We ask Master Gotama about the thing most helpful for hearing the Dhamma.”

“Giving ear is most helpful for hearing the Dhamma, Bhāradvāja. If one does not give ear, one will not hear the Dhamma; but because one gives ear, one hears the Dhamma. That is why giving ear is most helpful for hearing the Dhamma.”

“But what, Master Gotama, is most helpful for giving ear? We ask Master Gotama about the thing most helpful for giving ear.”

“Paying respect is most helpful for giving ear, Bhāradvāja. If one does not pay respect, one will not give ear; but because one pays respect, one gives ear. That is why paying respect is most helpful for giving ear.”

“But what, Master Gotama, is most helpful for paying respect? We ask Master Gotama about the thing most helpful for paying respect.”

“Visiting is most helpful for paying respect, Bhāradvāja. If one does not visit a teacher, one will not pay respect to him; but because one visits a teacher, one pays respect to him. That is why visiting is most helpful for paying respect.”

“But what, Master Gotama, is most helpful for visiting? We ask Master Gotama about the thing most helpful for visiting.”

“Faith is most helpful for visiting, Bhāradvāja. If faith in a teacher does not arise, one will not visit him; but because faith in a teacher arises, one visits him. That is why faith is most helpful for visiting.”
MN 95