Chains are cool, but are heavy as f. Side loading a hook is a huge no no as well. I know it's done all day every day, just saying tho. I've used a ton of nylon straps for heavy rigging, never had a problem but they were always fully tightened by hand on a smooth pipe surface or shackle rather than a tree which would grab like hell and likely make it choke up under tension. When the weights go up the tolerability of doing stuff wrong goes down, so proper rigging practices with shackles on loops, proper d sized stuff, no shock loading, etc can really make a huge difference. That being said, wire rope slings are about the best bang for the buck if the surface can tolerate it, and on trees you don't care so i would think that they would be ideal.
If you are having problems with them getting all kinked, maybe look into getting more flexible ones, with 7x36 construction rather than 7x19 if that's what you're getting, or maybe fiber core rather than wire, or maybe even some gator flex kind where they make them from really small wires so they're super flexible. The size of shackle/ connection can make a huge difference too, ever seen how they curl up ribbon when wrapping a present? They take a sharp edge of a knife and rub it on the ribbon, which stretches one side making it curl. The diameter of the connection (the d measurement) is what measures how sharp that is, so try to use more rounded (aka bigger) stuff. The choker eye connections are about the best, or maybe switch to an endless grommet so it uses a smaller diameter rope and you aren't choking it in the same place everytime.
Wire rope is safer than chain which is why it replaced it in the first place, 1 weak link in a chain and it fails but wire rope will show the condition far easier on inspection. Having said that tho, chain tolerates abuse, i spent all week rigging pipe down with chain falls and boiler hitches, the whole time cringing about the abuse it does to the hook but the stuff was slammed up against the roof and we're babying it way under wll (I'm talking like 1/3 of the wll, so 15 to 1 at most). The thing i really do like about chains is that you can get them with the adjustable grab hook master link thing, so you can have bridles that are adjustable so you can balance stuff, i even got some for my winch thing i like them so much lol.