The RG procedure can give Tc itself. See, for example, section 9.6 in https://www.amazon.com/dp/0201554097/?tag=pfamazon01-20. He does an RG calculation of the 2d Ising model and finds ##J/k_B T_c = (1/4) \ln (1+2\sqrt{2}) \approx 0.34##, compared to Onsager's exact result ##J/k_B T_c = (1/4) \ln 3 \approx 0.27##.I don't think the RG can give the absolute Tc, and it's usually done relative to the critical temperature T-Tc.
It will give you an approximation to the critical point and the critical exponents. How good an approximation that it depends on how good the approximations you made in your RG procedure are. See, for example, the section of Goldenfeld quoted above.i know this.
but the problem is, whether this approach can give the correct critical point
What exactly are you wanting to compute? If you want to find critical exponents and transition temperatures, RG will do that for you, and as long as you pick a half-decent RG procedure, the results should be half-decent. Obviously if you want to compare the results to experiments you are going to have to do some hard work to improve the numerical accuracy, but one can always come up with more refined RG schemes to that end.i am not satisfied with RG, because in most cases, you run into a non-controlled approximation.
RG is more an idea than a practical computational tool