Back when I hunted the NoCal Grasslands, you needed a bike. Not so much for the "make it easier", but for the "I gotta' beat those other six guys to 'my' spot'. Several things I learned:
Cheap bikes don't give problems with things coming loose, they give problems with the derailer. I.e. after about two uses, you'll have problems going into and staying in the gear you want. This can be a real PITA. The brakes never seem to be in adjustment (no big deal, really).
Mud, with any bike, is a MAJOR PITA. Put the waders in the trailer/box while riding...it saves sweat and saves waders. Bike trailers can be great, until you get on the side of a slope or one of the trailer wheels drops in a big hole...then they are PITA. I never had much trouble with flats, but many guys I know do.
Basically, bikes are a PITA unless you need the speed. If speed is of the essence, nothing else will compare. Otherwise, a push cart and walking is simpler and less trouble in the long run.
If you're going for a $100 bike, get a single speed. 10- or 15-speed units are fine if you're spending the price of anew B-gun.
If you've ridden both cheap mountain bikes and expensive mountain bikes, you'll quickly learn that the major difference is not that 1 pound frame weight, it's the quality of the components: the wheels, the bearings, the crank, and crank pedals, the derailer and chain, the levers, the cables....all the little things you don't think about until you have to use them for a season.