The monarch appoints "preformateurs" and "formateur" who are elections start talks between people from the different political parties and (try to) form a cabinet, in different stages of the process. That's where he (or possibly she in a few decades, when Filip dies or abdicates and Elisabeth becomes Belgian's first Queen, if that's still a thing then) has by far the most influence in politics.
Some people he shouldn't even have that influence, and I'm inclined to agree. In any case, finding another process for that is probably the most difficult issue to deal with in case we would transform to a pure parliamentary democracy.
Reserve power ... I don't know. I don't think he has enough power for that. Maybe symbolically, maybe that counts for more than I think. Some say the king and/or the whole royal family is one of the few things that still hold Belgium more or less together.
If indeed having a figurehead can protect us from civil war and other kinds of chaos, then by all means, let us have a figurehead.
In Brazil in the 19th century liberal democrats thought a monarch was unnecessary, after 67 years of successful politics under a king. Upon abolition the country devolved into a long succession of military coups and dictatorial governments.
There have obviously been successful countries without monarchy, but it is very risky to say that a figurehead does nothing. Reserve powers are a pretty real restraint.