What would happen if everyone became a monk or nun?

It is not possible every person to become a monk or nun. It is same as not every person become a doctor or nurse. Only few people have the faith and energy to become a monk or nun.

A layperson does not necessarily have to be a householder. A layperson is not necessarily clinging to a house if they live in a house. One can hold something but still not cling to it. When I use the word "hold" here, I mean to "have something" or "feel something" or "use something" without clinging to it, like for instance, "one can have love for a parent without clinging to the parent".

The distinguishing principle in the quote from MN 71 is:

'without abandoning the fetter of householdership'

I would assume (I will research later) 'the fetter' of householdship refers to concerns about & bondage to family, etc, as written in many suttas with the stock phrase:

...dwelling in a home crowded with children, enjoying Kasian sandalwood, wearing garments, scents and unguents, receiving good & silver... AN 8.54; Ud 6.2; SN 55.7; SN 55.53

A layperson can abandon 'the fetter' of householdership, as explained in the Haliddakani Sutta, which describes the 'household life' is a state of mind rather than a state of physical habitation:

The property of form, householder, is the home of consciousness. When consciousness is in bondage through passion to the property of form, it is said to be living at home. The property of feeling... perception... fabrication is the home of consciousness. When consciousness is in bondage through passion to the property of fabrication, it is said to be dwelling at home.

"And how does one not live at home? Any desire, passion, delight, craving, any attachments, clingings, fixations of awareness, biases, or obsessions with regard to the property of form: these the Tathagata has abandoned, their root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Therefore the Tathagata is said to be not dwelling at home.

"Any desire, passion, delight, craving, any attachments, clingings, fixations of awareness, biases or obsessions with regard to the property of feeling... perception... fabrication...

"Any desire, passion, delight, craving, any attachments, clingings, fixations of awareness, biases or obsessions with regard to the property of consciousness: these the Tathagata has abandoned, their root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Therefore the Tathagata is said to be not dwelling at home.

"And how does one live with society? One who is in bondage to the distraction of the society of form-impressions is said to be living in society. One who is in bondage to the distraction of the society of sound-impressions... aroma-impressions... flavor-impressions... tactile-sensation-impressions... idea-impressions is said to be living in society. This is how one lives with society.

"And how does one live free from society? The Tathagata has abandoned bondage to the distraction of the society of form-impressions, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Therefore the Tathagata is said to be living free from society.

"The Tathagata has abandoned bondage to the distraction of the society of sound-impressions... aroma-impressions... flavor-impressions... tactile-sensation-impressions... idea-impressions, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Therefore the Tathagata is said to be living free from society.

"And how is one intimate in villages? There is the case where a certain person lives entangled with householders. Delighting together with them, sorrowing together with them, happy when they are happy, pained when they are pained, he takes on any of their arisen business affairs as his own duty. This is how one is intimate in villages.

"And how is one not intimate in villages? There is the case where a monk lives unentangled with householders. Not delighting together with them, not sorrowing together with them, not happy when they are happy, not pained when they are pained, he does not take on any of their arisen business affairs as his own duty. This is how one is not intimate in villages.

Therefore, a layperson can end suffering (generally regarded to the degree of a Non-Returner).