For TV, and I assume it's the same for cable internet, we always say in the store that every connector (wall socket, splitter, extension connector, etc) is a point where the signal weakens and an extra point where problems can occur.
So, yes, I would remove the splitter and connect the router/modem directly to the cable without a splitter.
We recently had cable internet installed and it wasn't working properly after a couple of weeks. We replaced the splitter the cable guy put in with a cheapo one we sell in our store and the problem was solved.
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