That's my only experience of it. The nuts are usually made of a very hard plastic-like material (originally they'd be made of substances like carved cow bone) and not at all flexible.

The baking soda makes the superglue set almost immediately and also acts as a filler to give it some bulk, that's all. I'd have no faith in it to fix anything flexible like a plastic chair. The superglue most likely wouldn't stick well enough, and the join would be too stiff to be durable on its own.

The best I can offer is to suggest a method using fibreglass as used in car body repairs etc. (either the liquid filler on its own or backed up with fibreglass mat for reinforcement), possibly coupled with the other fixing methods you've tried, though that's also unlkely to be flexible and adherent enough to make anything like a permanent repair unless you're very lucky.

The only other method I can suggest is some type of plastic weld, melting the edges together. I've seen some folks on YouTube use plastic zip ties fed into a hot glue gun as a filler after using a hot blade to try to blend the edges of the crack together beforehand. That seems plausible, but the problem is that any thermoplastic when heated tends to go more rigid than is desirable and is unlikely to match the flexibility of the original material.

I also tend to try to fix stuff rather than buy replacements myself, but I think this is one case where I'd say the chairs have had a useful life and it's time to replace them, hopefully being able to send the old ones to be recycled into something new and useful again.

works good. still have a couple old chairs that I fixed that way. when it is just the top of the backs there isn't that much stress. when the handles or legs break then it is time to go bye. though some will work with a 5 gallon bucket as a spare leg in an emergency.

That bakingsoda cyanocrylate mix.
Might work with some reinforcement, but I suspect
The suggestion of fiberglass might be the best solution for durability.

Heat is also quite useful with some plastics.
Embedding some heated coat-hanger gauge wire might
be a good approach. Always depends on the type of plastic.

Finish it off with a torch-
https://imgur.com/gallery/v3dXbUG

I try to stick with white chairs so the sun damage doesn't show as bad.

might even have a tube of glue around

so even if you have to repeat it every season, you won't be out much.

Fiber Fix Tape
is a fiberglass like bonding glue tape.

I'll have to give it a try.