I've done this in my former pond (turned into horse pasture) for several years. It was swampy where the water "meandered" through, now my tire tracks have turned it into a channel/stream and one can walk up to the edge of channel and jump over it instead of sinking in mud within 20'.
Downstream (and actually on neighbors land) I did the same thing. He was building ATV trails so I "helped him" (as it helps me drain my land) and cleared all the brush from the stream/marsh bed. Just having 6" deep ruts really helps keep the water flowing. My city-slicker brother-in-law came to visit and wanted to ride the ATV so I sent him to "clear the channel", cause it's the muddiest place and most fun. He did a good job. Totally trashed (in mud) my 4 wheeler. But turned off "the stream" track too early and now water is going where it forms a marsh area. No problem to fix come this spring.
Here's first time driving through. The "channel" area starts around 0:40 secs. Channel is a lot more defined now after multiple trips.
First ride through - YouTube
I thought about buying an old "horse drawn" plow and towing it behind the 4 wheeler, or using my logging winch to pull it from 165' away through the marsh. Like you said, some of these places you can barely walk through without it sucking the boots off your feet, and can't drive through (except ATV). But so far just making tracks works