Why isn't pressure filtration popular compared to vacuum filtration?
It's much easier to assemble a system that's safe under external pressure than one that's safe under internal pressure. Convex flasks and tubes resist external pressure better than internal pressure. If something leaks or breaks, you have air rushing into the system, rather than reactants and products blasting out. For that matter, if you've put things together with ground glass joints or rubber stoppers or corks, internal vacuum (external pressure) works to keep them together, while pressurizing the system's interior tends to force them apart.

Also note that a vacuum system will never be under more than one atmosphere of external pressure. With a pressurized system, you need a way to limit the pressure to safe levels, and you need to make sure you know what those levels are.