Coffee

Side question, what's your roasting process?

On topic: As others have said. Investing in a proper grinder may be the next step instead of an espresso machine.

I agree with some other comments here that $100 won’t get you what you’re hoping for.

A better upgrade would be to start saving for a good handheld burr grinder (like an 1zpresso model) to replace the blade grinder. Your French Press and AeroPress will likely taste notably better!

If you do want to venture into good espresso, as someone else mentioned, start looking into Cafelat Robot or Flair. Both can make fantastic espresso at higher quality than semi-automatic machines of the same price point and require much less upkeep. But, for any espresso, you will need more gear (scale, tamper, vessels, etc.) which adds to the price. And all espresso machines will massively benefit from a good grinder.

Good luck!

french press or gaggiuino.

What about a moka pot?

Without trying to be gate-keepy about espresso, $100 is really too cheap. Since you just have a blade grinder, you'd have to get your coffee ground at the store/coffee shop. That makes it difficult to get the pressure/flow rate right without a pressurized basket.

If I were you, I'd stick to trying to make "espresso-style" drinks (especially if you want milk drinks) with your aeropress, or maybe getting a cheap mokapot, and if you want to spend some money to improve your coffee, spend it on a burr grinder.

If you are set on trying to get up and running with "true" espresso, look for used manual lever machines like a Flair or a Rok, and get coffee ground at the store, and get a pressurized basket.

I have a Breville Bambino, which I think is the low end of acceptable, but it's 300 USD new.

Honestly, you won't find any electric espresso machine at that 100$ price point that's capable of producing decent espresso, and they're usually a pain to use. They are plastic, disposable and cobbled together with the absolute cheapest parts as possible. Heating is bad, pressure is all over the place and build quality is inexistant.

Your best bet at that price point for real espresso would probably be a (used) Flair or Cafelat Robot. Of course these come with their own workflow and caveats and they are hard to master, but short of spending at least 5 times your budget on a well maintained, second-hand mid-range Gaggia or Lelit or equivalent, you won't find anything remotely as capable at producing real espresso.

I'm not sure that a $100 espresso machine even exists.

My humble opinion is you'll be hard pressed to find something that's less work than an aeropress, let alone anything that takes even less space than that.
If you're just looking to froth milk, maybe just get an electric milk frother.

Look for a $500 espresso machine that's used in selling for $200 or less

And you'll also need a grinder that can make espresso fine grounds. Your blade grinder won't work for that.

Not sure about $100. That's a push.

I have a cheap (but not that cheap) Delonghi Dedica which is a bit plasticy in places, but makes pretty good coffee. We bought it because it's narrow and fits on our limited kitchen top. It's about £180 if you can go higher.

For me, it was a huge step up from a mocha pot. Despite trying all the tricks, for me a mocha pot always made bitter coffee.

Have you tried second hand?

I’m not an espresso drinker but the biggest difference between cheap crap and cheap decent is whether or not it uses a pump to generate pressure or they cheaped out and just use boiler steam pressure.

Maybe a used flair classic, if you can live with manual lever brewing. I love mine, but you have to like making espresso and not just drinking it.

Edit: you'll need a proper grinder no matter which machine you choose.