change derailleur and shifter in an old road bike reconverted to hybrid
It's best to have chain, shifters that match your cassette, both in terms of number of speeds and make (and to a lesser extent model).

In a non-matching configuration, your chain will try to sit slightly between gears in certain settings, which is actually rather annoying.

Having said that, what you propose will work for locomotion, and depending on how much annoyance you're cool with bearing in a given day it may work well enough for your desires.

I tell you these things as someone who has piecemeal upgraged from 7 to 9 speeds; it isn't an ideal solution.