I was under the impression that the ZV-E10 had the same stabilization as the ZV-1. Something like off/normal/active.
OP: I’d suggest watching some YT videos on suggested setups, etc. Also you might want to consider a gimbal.
I watched the Gerald Undone video on the ZV-E10 (I recommend Gerald Undone if you want to learn about cameras and such). He shows the menu options in the video and the options are something like Off/Digital/OSS.
The Digital mode results in a slight crop in on the lens as the CPU will stabilize the video frame by moving the frame around inside of that crop to compensate for jitter/bumps of the camera. The crop in isn't ideal if you already want a rather wide 16mm sigma lens for example, but at this price point, it's quite amazing (Sure we'd all love to buy a pair of Sony A7Siii's with a pair of $1500 lenses - but for what the ZV-E10 gives at well under a thousand dollars, its pretty amazing). Keep in mind the ZV-E10 is even cheaper than an A6000 today (which is cheaper than an A6400).
The OSS mode is obviously dependent on a lens that support OSS. The difference between the ZV1 and ZV-E10 is that the E10 gives you changeable lenses! So now you could go buy a fancy optical stabilized lens like the Sony 10-18mm which will give you quite a lot on such a low cost camera. Personally, I'd take the changeable lenses on the E10 anyday. So many people didn't find the ZV1 wide enough and their are after market addons but not quite what you can get with good lenses.
Also for what its worth, I first bought some kit lenses and cheaper Sigma lenses for my Sony A6400 and A6000. Then later on I ponied up the big bucks and bought a Sony G-master SEL1655G and WOW the autofocus is sooooo amazing. It's fast and quiet and stable when focusing. There is very little or no magnification around the edges while autofocusing. So if you're on a budget buying a camera with a kit lens that you can then go and upgrade lenses later when budget allows to either the Sony 10-18mm or Sony Gmaster 16-55mm is totally worth it. That means choosing between the Sony A6400 and ZV-E10 today.
Of course this is all somewhat opinionated... but that's my experience and thoughts! I know the ZV-1 is still a great compact light-weight portable camera and is still great for those reasons. I just happen to have been quite happy with me lens upgrades.