I would call around to different bike shops. See if any of them let you rent bikes. And if you can apply that rental price towards the purchase of a bike. There will be a shop or two that will. Then rent a hybrid, road, and mountain bike. See which one you like best, and then buy from that category. A mountain bike or hybrid will definitely have low gearing. If you go road, make sure you have a compact crankset, with a big set of gears in the back(34 teeth little gear in front, 28+ teeth on the biggest gear in back.). Make sure you're comfortable pedaling while standing up.

FWIW, I just looked through my strava and I never went below 39/28(I have a 30/39/50 triple, so I could've) on rides similar to the stated commute. I'm not particularly fast, either. I'm a sit&spin guy, I don't really stand up unless I'm super tired.

I think 34/28 would be fine(which is what most road bikes will come with). Go 34/30 or 34/32 should be fine.

OP, I think just about any road bike would work, and $1000 is a generous budget. You can get a claris or sora spec'ed bike, which'll work just fine. Let the shops you visit know about your commute, and ask if you can get a medium or long cage rear derailleur with a big cassette. And that the bike has a compact crankset(34/50 teeth instead of 39/53, like explained earlier).

The biggest lesson I've learned about bike commuting is that I was way overthinking it and entirely too intimidated. Err on the side of small gears, find a way to carry your stuff that works for you. Make sure you have a couple blinky lights, front and rear. Maybe an actual headlight if you ride at night. Then just swing a leg over the bike and start pedaling. It's not too complicated. Don't overthink it.

When I went bike shopping I think I visited every bike shop in town trying to find the right one. Some shops I liked more than others. But with what I've learned now, all of them would've set me up on a bike that would've worked just fine.