I've now tried it by adding a bunch of random points and I find it very cool! It can make curves fit very snugly. Maybe enhance it by a mode that runs all the models and shows you which one has the least errors/best fit.

I don’t think you can do anything sensible here without making much stronger modelling assumptions. A vanilla non-parametric bootstrap is only valid under a very specific generative story: IID sampling from a population. Many (most?) curve-fitting problems won't satisfy that.

For example, suppose you measure the decay of a radioactive source at fixed times t = 0,1,2,... and fit y = A e^{-kt}. The only randomness is small measurement error with, say, SD = 0.5. The bootstrap sees the huge spread in the y-values that comes from the deterministic decay curve itself, not from noise. It interprets that structural variation as sampling variability and you end up with absurdly wide bootstrap confidence intervals that have nothing to do with the actual uncertainty in the experiment.