Yeah, on the whole, announcement mics should have no effects at all, even compression (or it will tend to feed back more than an uncompressed mic).

The only problem putting an announcement mic direct into a sub and stick PA is being able to quickly adjust the gain or turn it off if it starts to squeal. It’s not often easy to bend down and adjust on the sub if it’s behind you or anywhere out of reach. I would recommend the little mixer just for the announcement mic (if you normally run your vocal mic into your keyboard). Plus, you can get away with a cheaper mic (can’t tell you how often I’ve seen an announcement mic dropped or banged about!) if you have an EQ on the mixer.