I'll give you an answer from a historical linguistic perspective, which hopefully is okay, considering your question is about language.

There is a false premise in your question, in that pre-colonial inhabitants of the Philippines spoke Old Malay. They didn't. Old Malay was spoken natively around the core area of modern Indonesia and Malaysia, from Borneo and even beyond in the east to Sumatra in the west (namely the Malay-dominated Srivijaya mandalas). Malays had superb navigational skills, and traded far and wide: I can't speak for archaeology, but there are traces of language contact with Malay from Madagascar all the way to Manila (and maybe beyond into Taiwan).

Malay was certainly known in parts of the Philippines that participated in maritime trade, but it was never widely spoken natively. Different languages were dominant in various parts of the Philippines: Ilokano in northern Luzon, Tagalog in Maynila Bay, Cebuano/Binisaya in Central Philippines, Tausug in the Sultanate of Sulu. These languages have retained their prominence to this day, and are still used as local lingua francas, although the influence of Filipino has increased greatly over the years.

I think you may be confusing Old Malay with Proto-Malayo-Polynesian. Old Malay would be a language of the Malayic group, spoken in Borneo or perhaps other areas of insular SE Asia after migration from Borneo. The first Malay-speaking city-states in Sumatra appear in the 7th century CE, a mark of Malay naval dominance in SE Asia.

On the other hand, Proto-Malayo-Polynesian (PMP) was spoken by Austronesian peoples who migrated from Taiwan to Northern Luzon around 4000-4500 years ago. That stage is likely the lowest common ancestor of all Philippine languages: there have been attempts to put all Philippine languages in a subgroup, but they are very controversial and most definitely not the majority view. At around that time, PMP speakers invented the awesomeness that was the outrigger (essentially a log tied to a boat that keeps it from overturning at sea), and used their supreme naval skills to colonize most of insular SE Asia in short order.

So why Tagalog then? There may be different views why it was picked over, say, Cebuano, but the gist of it is due to the influence of the Maynila Bay on maritime trade (and perhaps also political influence as an aftermath of colonialism). It is important to note that Tagalog was far from unknown outside of its immediate area of influence: we can find old Tagalog loanwords in Tausug, at the southernmost end of the Philippines, so it's not like it was completely unheard of before becoming the basis for the national language.

tl;dr Old Malay has never been spoken natively in the Philippines. Filipinos tend to speak local lingua francas and/or Filipino, because it's always been this way (minus the national language).