Hello,
Like I said above, lsqrsolve is sometimes faster than classical (not least-squares dedicated) methods with good initial guesses, but it does not handle constraints on the parameters and leastsq does not handle other constraints than bounds. Moreover, comparing methods with a single initial guess does not test the global convergence properties.
To me, fmincon is a kind of swissknife for all kinds (with or without linear or nonlinear constraints) and sizes of problems and there will always be other methods showing better performance on specific and/or small size problems (see e.g. https://scilab.discourse.group/t/large-scale-nonlinear-optimization-with-fmincon-in-scilab-vs-matlab which highlights a use case out of reach of lsqrsolve or leastsq).
S.