I wonder how this statue could be dismantled or kept in disguise during the Yeltsin's years. It would be utterly dominating over the landscape and ideologically undesirable in the 90s.
Of course, the Palace itself can be used as host for some of the post-Soviet governing body, but the situation would look ridiculous.
It was supposed to host the Party congresses with the main hall capable to accommodate some huge numbers of people. Which created additional problems: people in that enormous hall had to be able to hear and see the leaders making speeches. I was told that 2 special research institutions had been created to resolve the issues of acoustics and visibility (we are talking 1930's and the huge TV screens are not yet available). Not sure that this monstrosity could be used for something meaningful except, if all the problems were resolved, as a concert hall (as Palace of the Congresses built during Khruschev's time). Dismantling of the statue would be an interesting engineering task. And quite expensive one.
IMO, public swimming pool was the best usage of the site because Soviet Moscow was desperately short of them but there was (surprise, surprise) a major drawback: the pool was within a spitting distance from the main Moscow art museum and with the pool being open one and functioning the whole year, moisture (in the winter you could see a big steam cloud over it) had been allegedly damaging the valuable paintings.
Well, now they rediscovered religion (as in figured out which hand to use when making a cross sign) and "rebuilt" the huge cathedral which was initially on the site. In a process, invented a legend regarding construction of the original one, and couple other related things. The people seemingly swallowed all the BS hook, line and sinker.