Strong disagreement.

By far the biggest impact on steering stability in a bicycle, and indeed in nearly any steered vehicle, is mechanical trail. This is the tendency of the front wheel to want to act like a caster, or put another way, its tendency to resist being steered away from straight. On a bike, various factors impact on mechanical trail, but assuming the steering axis is at a standard angle, the big one is wheel size: small wheels are much more twitchy than big wheels. The other big one, which could be a design flaw in his admittedly crummy bike, might be a high rake, which will also strongly impact on trail.

Wheelbase does have an effect on perceived steering stability but it's nothing close to the effect due to trail.

It's possible that the OP has other problems (headsets etc.) which should be looked at by a competent shop tech. But all other things being equal, yes, the effects he's describing are symptomatic of the kind of bike he bought, and its likely quality.