The first point seems like a genuine and substantial objection to the secretary problem's framing of the issue.

You've sort of addressed it by suggesting that maybe we should try a bunch of partners and then choose the next one that's better than 90% so far, rather than the best that's better than all so far, but it seems to me that once you've noticed that P is the wrong problem you should be trying to figure out the right problem and solve that, rather than solving P and then applying ad hoc tweaks to the answer.

I believe people have analysed variants of the secretary problem where, e.g., each candidate has a quality score (drawn independently from some known distribution, I guess) and you want to choose a candidate to maximize expected quality.

[EDITED to add: I see that anon85 has made similar points.]