Why don't I like them?
Why do I think they are lazy game design?
Why do I think it was a good idea that we generally as a hobby moved beyond them in the 90s?
Why do I think it's hideous recursionism and conservatism in spite of progress to use them?
Let's start with a few points that build my case...
(But first...what is a "hodge-podge" system? It's a game system made up of disparate components, where different parts of the system are designed to handle different narrative or simulatory effects in different ways. Think of thief/rogue skills in old D&D being based in percentages, while nothing much else uses percentages...or games where the combat system is completely different to everything else in the game.)
I don't think there should be obstacles preventing people from gaining entrance into our hobby. I similarly don't think there should be gatekeepers monitoring those obstacles. I don't care if those gatekeepers are grognards living in their mothers basements, hipsters protecting "the emperors new gamesystem", rebellious punks screaming their evangelism at the world, or anyone else for that matter. Hodge-Podge systems are inherently not approachable; in a diverse system a new player can learn a single system that roughly approximates what is necessary to drive the story, but in a hodge-podge system new players have to remember a variety of mechanism systems (even though each system may more carefully approximate the output). Forcing more learning is a barrier to entry, much the same as $100 books, on top of custom dice, cards and everything else that certain companies expect new hobbyists to acquire before their games can be played.
One of the other things I don't really like about hodge-podge systems is the fact that there is often a disparity in the ways things work with some systems compared to others. As an example, I look to the way rogues in early D&D used percentiles for their skills, while other classes used the d20 in various ways. This just bugs me, there's no real reason why one sort of mechanism needs to be used for a pass-fail result, while another mechanism is used to similarly generate a pass-fail result when a different skill set is used. Saying that such things are necessary is just claiming that the old things are good for the sake of tradition (which leads back to my first point, barriers for the sake of barriers).
A third point, this sort of harks back to my issues with the OSR. Hodge-podge systems often seem to be amalgams of stuff that is known to work, or more accurately they are known to work as individual components but not necessarily work as a coherent whole with one another. Those who like hodge-podge systems seem to consider this a virtue, it allows them to change gears during the telling of a story to change up the pace and keep things interesting, but this feels like a straw man argument against more diverse and robust universal mechanisms...and similarly feels like an excuse to avoid pushing the envelope to develop a good new core game mechanism.
I'm sure there'll be debate on this one...