When I became more environmentally conscious many years ago, I decided to switch to a safety razor to reduce plastic waste. I landed on a reputable safety razor with a 'mild' angle, something that would be good for sensitive skin. Even with a good shaving soap, I found that I would regularly still get some discomfort from razor burn, despite learning a solid technique.
Enter: the Henson. This was touted as a 'new' design of safety razor that fully supported the blade, making it particularly well suited for sensitive skin, and much harder to nick yourself.
Well, I've been using this thing for a few months now, and I can honestly say the marketing isn't an exaggeration, it's been a game-changer for me.
It really is MUCH harder to nick myself with this, to the point where I can press it against my skin firmly like a disposable, and I still don't get irritation. It's one of the most pleasant shaving experiences I've had, and I can heartily recommend one for anyone who wants a truly mild safety razor.
Its made of machined aluminum, and built to tight tolerances. I anticipate I'll be using this sucker quite literally for the rest of my life.
Though do bear in mind, If you have a thick beard or non-sensitive skin, the mild henson might not be ideal for you. As an example I still use my old razor with it's more exposed blade to shave my head, as this henson gets clogged up far too quickly for that application (unless the hair is already really short). But for the face with light facial hair? Perfection.
I've found the whole ritual aspect is pretty much unnecessary, and I kinda regret buying some of the accessories that people in shaving forums say is needed, like a lathering bowl (Honestly you can apply a fancy shaving cream directly to a wet bristle brush and it works just as well, and regular old barbasol has worked fine too).
A safety razor would work fine for your use case, but a henson especially, since they're pretty much just as easy and safe to use as a cartridge razor. Like others said, you can still cut yourself if you make sideways motions with the blade, but I think that's probably true for cartridge razors too.
If you want to get a budget razor just to see if it's something that would work for you, the Lord L6 is actually pretty well regarded, and only costs like 7 bucks.