No flips for you!
This is a pretty narrow view of games, one that seems very entrenched in a specific form of play. As an entry into the topic, it really fails to address how this kind of play can create the type of experience the OP has requested. You specifically reference the, "yes, and..." improv advice, but this counters character driven play because it asks you to subsume character to the play by agreeing with the situtation as presented by others and building onto it, whereas character play requires pushback on things that are central to or important to the character.You will never get a meaningful roleplaying experience worrying about rules. The moment you look at your sheet the roleplaying is over, so to achieve whatever character immersion you want, you have to go back to OD&D or D&D or a relaxed AD&D and just roleplay with a system that has next to no stats. It also helps if you have a DM that doesn't worry about rules either and just says "yes, and..." a lot and you say it right back to him. This is why 5e was such a letdown to me after 30 years away. I don't want to make a religious check to see if I know something. I want to approach an NPC and have a conversation with them and learn something, Keep on making tedious skill checks and you'll never have a character driven campaign. And throw away those backgrounds the books try to foist on you. I have to play a character four or five times, get the feel of him, and then I know what his background is. Yes I can act out playing whatever was randomly rolled for me, but it is certainly more immersive and natural to just play five sessions, realize my character detects magic a lot, and suddenly the answer comes to me and the character is fleshed out. Then I play accordingly.
Don't get me wrong, this is a perfectly fine way to play, and I'm happy that you enjoy it. It isn't, however, an actual solution to the OP except in the ad hoc, unpredictable way I've described previously. Sometimes lightning strikes doing this, but, like lightning, that's neither reliable or should be expected.