There's a lot of options out there. A lot of creators use the Sony a7, but it's far from the only option out there.
The most important question to answer is this: what will you be shooting? Because if you're just going to be vlogging, that ZV-E10L is probably going to be just fine. You might not need an interchangeable lens system at all, so a point-and-shoot style camera might be just fine. I would avoid DSLRs because they use mechanical shutters, which have some long term problems but in practical terms they're just one more thing to take up battery when you're recording video. If you're looking for interchangeable lenses, the word you're looking for is "mirrorless", though you might find other odd terms for it like "MILC".
This may be a bit dated at this point, but the cameras I heard of the most for people who did both video and photography were Panasonic's Lumix Micro Four-Thirds (MFT) cameras. I did a quick search and it looks like most of the kits (including at minimum the body and a basic lens) were about $1000 US, and some models were even less. Looking online I see a retailer who has a Lumix G9 packaged with a 12-60mm lens with OIS for US $1097.99. While the lens is of ho-hum image quality, it's more than good enough to get started with, especially for video (where the image stabilization will be important, even with a gimbal). You could probably get an even better price if you look on the used market.
Sony has an upcoming camera release (29 August). I think the rumor is that it will be a camera primarily for video shooters (see link below). My suspicion is that it will be beyond your budget, but I'm not well versed on Sony pricing.
You can also try to look at the used market, especially sites like keh.com and mpb.com
Any camera without interchangeable lenses is a trap for serious video creation. The lens isn’t just some thing you attach to your camera like a gimble is, it’s the most important piece of gear in determining how good your video quality is.
Regarding the a7iii... To cut a long post short, full frame cameras are way better for photography but the extra cost means you need to up your budget beyond the a7iii to get comparable video features.
One aspect that wasn't that much explored in this topic is recording time and battery life.
You mentioned:
All I want is to record however long I feel like (though realistically always less than 1 hour continuously) without issues or worries like overheating and know I'll have a good video afterwards.
Heat: I don't think it's a problem anymore? At least I know my X-T3 can record fine (in slow motion no less) as long as the battery holds out.
Legal time limit: for taxation reason, most camera sold as, well, camera have a 30min recording limit. This can by annoying, but 30 minutes for vlog is a long time, and you're certainly going to cut the video anyway.
Battery: I know that a friend's camcorder (Sony AX53) can record continuously for 2 hours on a single battery, while my X-T3 will use two batteries. Try to get some info about how long a battery can last (and buy at least two of them anyway). And it's 2023, so try to look if you can plug your camera a powerbank and record at the same time.
There's a lot of options out there. A lot of creators use the Sony a7, but it's far from the only option out there.
The most important question to answer is this: what will you be shooting? Because if you're just going to be vlogging, that ZV-E10L is probably going to be just fine. You might not need an interchangeable lens system at all, so a point-and-shoot style camera might be just fine. I would avoid DSLRs because they use mechanical shutters, which have some long term problems but in practical terms they're just one more thing to take up battery when you're recording video. If you're looking for interchangeable lenses, the word you're looking for is "mirrorless", though you might find other odd terms for it like "MILC".
This may be a bit dated at this point, but the cameras I heard of the most for people who did both video and photography were Panasonic's Lumix Micro Four-Thirds (MFT) cameras. I did a quick search and it looks like most of the kits (including at minimum the body and a basic lens) were about $1000 US, and some models were even less. Looking online I see a retailer who has a Lumix G9 packaged with a 12-60mm lens with OIS for US $1097.99. While the lens is of ho-hum image quality, it's more than good enough to get started with, especially for video (where the image stabilization will be important, even with a gimbal). You could probably get an even better price if you look on the used market.
Sony has an upcoming camera release (29 August). I think the rumor is that it will be a camera primarily for video shooters (see link below). My suspicion is that it will be beyond your budget, but I'm not well versed on Sony pricing.
You can also try to look at the used market, especially sites like keh.com and mpb.com
https://www.notebookcheck.net/Sony-teases-likely-A7C-II-and-A7C-R-launch-date-with-August-29-New-product-announcement-stream-page-on-YouTube.742991.0.html
Irrelevant to your use case, but I went into photography three years ago, bought a cheap Canon DSLR and got right into it.
DSLR cameras may not be good for your use case (4k video at 60FPS.)
Budget answer Sony a6400.
Upgrade options Sony A6700 (released this month).
Panasonic is great for video but still far behind Sony in autofocus capabilities.
I am not sure about latest Canon R series which might be equivalent of the above Sony cameras.
Any camera without interchangeable lenses is a trap for serious video creation. The lens isn’t just some thing you attach to your camera like a gimble is, it’s the most important piece of gear in determining how good your video quality is.
Regarding the a7iii... To cut a long post short, full frame cameras are way better for photography but the extra cost means you need to up your budget beyond the a7iii to get comparable video features.
For video, Canon 5d mk 3 or 4, Lumix GH4 or 5 are easy bodies to recommend. Picking the lenses will take the most money. Go used.
One aspect that wasn't that much explored in this topic is recording time and battery life.
You mentioned: