So as a little update there's a straightforward solution to the weighted pushups problem if you're willing to shell out the money.
https://kensuifitness.com/products/kensui-ez-vest
The primary feature here is that you can actually load the vest with olympic (or even standard with an adapter) plates, which makes it wildly more versatile than most vests. It's loaded such that the weight rests on the thoracic spine and should remove any concern about lower back issues. I wound up buying one of these after a long string of coming up with my own solutions using dip handles and climbing vests - I may even write something up at some point as it was kind of a fun journey.
But yah, as an easy solution, this is it. All you really need is a pair of pushup handles and something to elevate your feet on that are close in height to the handles and the dimensions of the vest are such that you get a full ROM with a consistent depth. Here's a video of me using it:
As a note the dip belt version is a little easier, likely because the load is further away from the shoulders, whereas this is noticeably a bit harder. In a pinch I'd still try the dip belt version to see if you like the movement, and if you have a lot of disposable income to throw around, the Kensui EZ Vest really is a pretty solid solution. The vest should work well for other stuff, off the top of my head including hands-free bulgarian split squats, inverted rows as well as the usual weighted chins and dips.
If anyone is actually interested in buying the thing let me know and I can show you some tips to knock some money off that I discovered poking around the website. Basically if you hang out on the website it'll pop up a little survey which gives you like 5% off with a code (mine was just "SURVEY" which maybe is universal?), and then they have the option of getting some accessories for a discount if you order the base vest (I got the longer bumper plate pegs so I have as much space as possible, particularly for the rear-loaded weight). I got the max (loadable up to 225 lbs), which I think is the way to go if you're going to shell out this much money so you can use it for as many things as possible.