Check out little known Peaking Current Source, also known as "Wyatt Current Source".
https://www.edn.com/peaking-current-source-has-high-rejection/
That's an interesting circuit, thank you for posting the analysis.
Unfortunately, its output is (ideally e times, apparently) lower than the input, so not great for driving heavy loads. But the independence of input current is pleasantly surprising.
I have seen similar things in some TL431 implementations, but off the top of my head I don't think they made use of this output current stability. On the contrary, when your circuit is biased with a PTAT current, it looks like the output will simply be a PTAT current too at some fixed ratio to the input.
I noticed that OP mostly tries to exploit the dependence of Ic on Vbe. This is not very reliable in discrete implementation because of thermal drift: 1°C of heating on the current generating transistor has the same effect as increasing its Vbe by 2mV, which increases Ic by about 6%. Current effectively doubles at 10°C temperature increase, for the same applied Vbe.
You may actually get more stability by feeding a constant base current into the transistor. β also increases with temperature, but it takes some 100°C difference before it is doubled

The circuits posted by Benta could be better because they sense load current in resistors. Perhaps a Wilson mirror would be decent (the output "cascode" transistor takes all the load-dependent heat). ICs have an advantage of being able to match temperature of multiple transistors to a fraction of a degree.