Why do UK border officers question EEA nationals entering?

As a retired Immigration Officer, there is a madness to questioning people. Most people do not realize that you were being watched well before you enter the inspection station area. You may notice that most officers have a ear piece or radio. You are being observed and whoever is traveling with you as well. The officer is going to ask redundant questions. But you’d be surprised how many people screw that up. Immigration officers knows that they’re being lied to most of the time. But you get caught when the stories don’t add up. Lots of the time they will ask the same question twice, once at the beginning and then at the end. It is hard to remember your lies when you are under stress. You have to remember that you have limited rights when entering any country. Even if you are a citizen of the country.
Once it’s determined that you are a citizen of that country, the inspection is over. But it is just not having the passport that proves it, because of fraud in passports. The officer makes that determination.

The other answers have given good general reasons. I want to add a little bit to your particular case, though. In OP you mention that you asked the border officer to stamp a piece of paper that you had brought for him. You further clarified in a comment:

A blank A4 sheet torn in half which I sometimes bring as a souvenir
stamp sheet. When approaching each (French+British) officer, I said
“Good evening”, handed over my ID+sheet and said (in the respective
language) “could you stamp this sheet for me please?”. The French
officer glanced at the ID for 2 seconds and stamped without uttering a
word, while the UK officer briefly questioned me before doing it

There is really nothing wrong with your stamp collection hobby, but it may be one of the reasons the border official struck up a conversation with you. He might not even usually ask EU nationals anything at all. You did something unusual (something less than 1% of travelers do.) Therefore, you singled yourself out, and got his attention. It’s a normal human reaction to engage in a small bit of conversation whenever someone gets our attention. Of course, he is a border official, so the questions he asked were totally professional and appropriate. But if you hadn’t presented the A4 sheet and asked for a stamp, there might have been no questions. Here’s some related reasoning from another answer.

The other answers are also good, by the way. I’m just trying to add another angle. And there’s nothing wrong with your hobby, either, I’m just saying the officer’s questioning could be related to it.

Border Officers are trained to look for anything suspicious, and rightly so. Their job is not to welcome you to the country, but to protect the border.

Just because your nationality grants legal right of entry there could be other circumstances that affect your entry; you may be smuggling, your docs may be forged, you may have other nefarious intentions.

An officer will sometimes question you to help discern whether you are legit or not. If you are not then you might be nervous or give other signs. Its just one tool they have in their job. For what its worth, I’m a British citizen and I often get asked questions when returning from abroad (where have you been, where did you stay, how long were you away, was the weather nice, etc). I just give calm answers and don’t stress. If I had a couple of kilos of hard drugs in my bags I’d probably be sweating and wide eyed and the questioning would reveal my discomfort.

My experience at the UK border is the same and I even have had a short questioning when leaving my own country. Something I am definitely allowed to do. At some point this was at a smaller airport where you have to go trough the border control to reach a few extra-Schengen gates.

The border guard asked where I was flying. I considered this to be a silly question considering that it was written clearly on my boarding pass and that there were no other flights in that extra-Schengen part scheduled for that day. As I was tired after a long day, my sentiment was clear from my response. The border guard picked up on this and replied that he didn’t really care about the answer but just wanted to hear me talk to make sure that I was who my ID card said that I was. We’ve had a short chat and I’ve apologized for my less than diplomatic reaction.

My conclusion from this is that they don’t want to establish if you are allowed into the country, but rather try to catch some deceit. Are you really the person in the ID document? Do you act as someone who has something to hide? They don’t really care about what you say, but are merely looking for clues that a person is suspiciously nervous, lying or trying to hide something. As long as you don’t admit to criminal activity they will be more interested in the way you say things than in the content of your answers.