Present Simple instead of Present Perfect
- American speakers tend to use the present perfect much less than British speakers.
- Lately I get the feeling that... is just idiomatic AmE. It's not the same as, say, Lately I've been feeling sick ..., which is normal AmE: it's present perfect progressive because it's a consistent feeling. I use the present perfect much more than most AmE speakers, but I wouldn't say Lately I've had the feeling that... unless I didn't have it at the moment of speaking. Lately I've got the feeling... doesn't feel like AmE to me.
- ... have you noticed how much more time Paul has {spent/been spending}...? is idiomatic AmE, but I suspect that a lot of AmE speakers would say it in simple past: ... did you notice how much more time Paul {spends / has spent / has been spending}...?
The sequence of tenses is difficult for all speakers, native speakers included. What each tense, aspect, mood, voice, etc. is used for is a mystery to most. We just instinctively say what we say and use whatever tense seems to fit. IOW, I don't think it's possible to provide a rule-governed explanation of why people use one tense instead of another.
I agree with Reg Dwight that "there is no rule that 'lately invites the Present Perfect'." I'd say "Lately I feel sad about how vitriolic American politics has become" as soon as I'd say "Lately I've {felt / been feeling} sad about how vitriolic American politics has become". It'd all depend on whether I was feeling sad at the moment or whether I'd been riding a roller coaster of sadness and indifference.
There is no rule that "lately invites the Present Perfect". In fact, with Present Perfect the sentence would mean something different.
- "Lately I've got" — you got it once, and it's a completed action. (Whence the very name, "perfect".)
- "Lately I get" — you keep getting it, and there's no end in sight.
Likewise with the second sentence, "Paul has spent" vs. "Paul spends".