I own an old early 2000s hybrid Gary Fisher with straight bars, large tires (smooth down the middle, quite knobby on the sides). I also own two very fast and expensive road bikes, deep carbon wheels, fast aero position, the whole nine yards.

The difference in speed is tremendous. It easily takes the same power to average 15mph on the hybrid that it does to average 20mph on the race bikes.

Now, if the tires were practically the same, say tubeless conti gp5000s 28mm on the hybrid and the race bike, I'd put the difference much closer, probably between 18mph and 20mph. So yes, in my experience having these two types of bikes and actually testing speed at similar power outputs between them, I'd certainly say it would improve your commute, but the difference in wheels would probably have the largest contribution (so, if you just upgraded your wheels, perhaps a better investment?).

As for drop bars, I really only use them on fast, sketchy, descents, cornering at speed, drafting in a race or A-Ride, or smashing a Strava KOM. If I were commuting by bike, I probably wouldn't touch them and wouldn't really have that impact my buying decisions too much.

As for a CX bike, I personally would immediately say no, specifically because of the gearing, and secondly the twitchy handling. 'Cross is highly technical so the bike is really designed more for quick turns and slow speeds on crazy terrain, if you were descending or have a good tailwind on your commute there's a good chance you'd be spinning out.

A Gravel bike is like a CX bike but has a larger range of gears as it is often expected to go near tarmac speeds so there wouldn't be too much of a worry of spinning out, the geometry is often a bit more relaxed than a CX bike too.

That being said, I would simply recommend a aluminum endurance bike, if I were buying in your situation I would probably get one of the following:
Domane AL 3 $1,019.99
SYNAPSE DISC SORA $1150 (yes it has disc, but mechanical disc isn't particularly better than regular rim brakes, the integrated cable routing is nice though).

Grade Elite $1000 (value for days...)
Endurace AL Disc 7.0 $1599 more expensive option but it has basically everything, I doubt you'd be riding this around and feel like you really need to upgrade anything.

I would also throw in a few recommendations for flat bar bikes that are fast and would get the job done, but, aside from that one I mentioned, I really don't ride them and would never actually buy one.