Oh, I like present tense quite a bit (the last novel in present tense I read was a historical, Wolf Hall by Hillary Mantel and I loved it to pieces), but maybe somewhat less so for first-person narration. (Hilary Mantel used very close 3rd limited for Wolf Hall).
Problem is, first person present tense can feel like someone is doing a running commentary on everything they do, which can feel pretty artificial and actually undermine the immediacy one wants to achieve with present tense, reason being that commenting already involves a certain level of detachment of the commenter and a certain regard towards an audience. You are either caught up in the moment or commenting on it. There's a natural gulf between narrating I and experiencing I, at least if the experience is properly engaging. Using past tense acknowledges that and can actually feel more natural.
Of course that gulf is sometimes bridged when we get so caught up in our own story that we relive the past experience but it can be difficult to maintain that level of excitement over the entire novel. And in the quieter passages that sense of immediacy can feel forced.
You don't have this problem if you do a proper interior monologue, something like Schnitzler's Leutnant Gustl for instance.
How long is this going to last? Better look at the watch. Or maybe not, it's not proper behaviour for such a serious concert. But who's going to see? If someone sees it they're not paying any attention either and I don't have to be embarrassed ...Only quarter to ten? ... It feels as if I've been sitting here for three hours.
This feels fairly natural; it's simply Gustl talking to himself. He's not describing things (something that would suggest awareness of an audience), and he's not narrating his actions either (I sit in the concert. I fret about. I look at my watch). There are no considerations at all of a potential audience, no explanations, no transitions between thoughts. You read
Oh no....."With the collaboration of the choral society" - choral society ... odd! I always thought that's something like the Viennes dancing singers, that is, of course I knew it's something different! Ah, memories....
and you have to figure out on your own that he's reading the programme now. There's no "I look into the programme. It tells me that the concert is held with the collobarition of the choral society."
As a result the whole thing can be hard to follow. But it does feel right, and most of all, very immediate. Because that's the thing about immediacy - there can be no mediation, no filtering, no concessions to a reader. In short: no narration. It's great if done right, but pretty hard to pull off. And it demands quite a bit, not just from the writer, but also from the reader.
If you want immediacy, don't go for half-measures. You won't just need present tense, you'll need something as close to stream-of-consciousness as you dare to use without alienating your target readers. (Tolerance for stream of consciousness varies widely across readership).
It's not too different with 3-person limited present tense either. It works better and better the closer the perspective is tied to your focalizer character. But I do feel you get away with slightly more narration than with first person present tense.