What would be the best way to then remove the remaining ARCH effects? This is according to the Ljung box Q statistic of residuals squared and ARCH-LM test
First, the ARCH-LM and likely the Ljung-Box tests are not directly applicable on standardized residuals from a GARCH model as the null distributions of the tests statistics are different than the standard ones (those that apply for raw data rather than model residuals); see e.g. Wooldridge (1991) or Francq & Zakoian (2011), Section 8.4. What you could use instead is the Li-Mak test (Li & Mak, 1994); it was developed specifically for the standardized residuals of a GARCH model.
Second, if the remaining ARCH effects are genuine, try a different specification of the GARCH model: either change the lag order of the vanilla GARCH or try GJR-GARCH or other modification.
Is there even a need to remove all ARCH effects after the GARCH estimation (given GARCH models are iid), if one wishes to forecast volatility?
You would like to have the assumptions of the model satisfied. A GARCH model assumes the standardized residuals are i.i.d., so there should be no ARCH effects in them. Now if the violations are small and the model is sufficiently simple, it might still do well. There is a fine line between underfitting (simple model, some assumptions violated in sample) and overfitting (complicated model, all assumptions satisfied in sample).