You don't necessarily have to show input to every lake. What I would expect is for more of those lake basins to have outlets.
Sometimes it's sufficient to show navigable rivers, or also those broad enough to be serious barriers. But a dense net of watercourses could also be used to hint this area I've here is wetter than that river-scarce area over there. Too, to a practiced eye a river network says as much about the relief as do explicit topography symbols or contours.
In fact, your network may be saying things you don't intend. For instance, for a river to parallel a coast means there's higher ground in a coastal strip, keeping water from otherwise draining off the land quicker. And when a river runs a LONG way near the coast, think what that implies about how high the coastal barrier must be, far upstream (since upstream = uphill).
Over the ages, all but the deepest and driest basins would tend to fill with water, then their overflow would cut the low spot even lower, until impenetrable bedrock was reached, or the climate dried enough to make evaporation overrule rainfall. The bigger the watershed, the less likely there wouldn't be enough rain to EvEnTuAllY fill it in. Sure, there'll be some DeadSea/ GreatSaltLake situations. But there's also US Great Lakes and African Great Rift Valley chains of lakes. When I use Fractal Terrains to generate landmasses, there are always too many dead-end basins. FT lets you do the other likely real-world eventuality - even if there's not huge blind basins full-to-brim of water, over ages some of those would fill with sediment, making for nice plains, steppes, and such , with maybe room for only small lakes , whether dead-end or overflowing.
Apparently, since even a brand- new continent but recently arisen from the depths or volcanically piled up or magically emplaced has to have rainwater runoff paths, you can figure some rivers predate the mountains - and cut channels and outlets AS the mountains are pushed up. You can see those cases all over a world map. So some of those FT basins I fill with dirt, and others I cheat and tweak outlet channels into place to drain. Wilbur can do some such alterations on an initial terrain set, I've just never learned it.
All that to say - don't just scatter rivers to drain all the land - a good first aim! But also figure your general highs and lows, and where they would push and pull the water. You don't have to devise a full set of ice ages and plate tectonics and climate shifts - but a few generalities will go a long way toward heightening plausibility.
To directly answer your question, once I was satisfied with river placement, for lots of map types I would keep at least as many as you show. I might slim them down, and/or mute their color a bit. Is your original map file big enough to permit several different widths of watercourses, and /or tapering them?
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