Given the Suns gravitational field, and the orbit that Mercurius has, the speed of the planet can be determined. So the argument that a point mass would have the same precession does not dispute the argument that it is the relativistic mass of the point mass/planet that determines the precession.
I will not venture into arguments* whether grandparents explanation is ‘correct’, ‘best’ or ‘useful’, there is an equivalence between ‘gravitational field + orbit’ being the reason and ‘(relativistic) speed + orbit’ being the reason. * A sybling comment states that relativistic mass is avoided in modern physics. As someone who did physics 30 years ago, I can not deny or corroborate this statement. And indeed, the explanation of grandparent is not the way I myself think about the perihelium precession. But that does not make me certain enough to say it is ‘wrong’.
Edit: typo ‘of’ —> ‘or’