Relation between witness and problem statement
Consider a problem named 3SAT_NORM which is a version of 3SAT restricted to instances wherein the number of literals used is equal to the number of clauses. For 3SAT_NORM, every input of $n$ clauses requires the witness to give $n$ values for the satisfying assignment. Thus, you can't simply limit the size of the witness and relate the difficulty of the new problem -- the witness size is a function of the instance size.

But, you could conceive of a different witness than the usual one for 3SAT_NORM. You could take a convention where if the witness to 3SAT_NORM only defines some subset of the literals, then the other ones are automatically assumed by convention to be given an assignment of "true". 

Now, suddenly, we can limit the size of the witness and make observations about the relationship -- in particular, restricting our witness to be smaller is the same as restricting 3SAT_NORM to include only formula where satisfying assignments are guaranteed to have a lot of "true" values.

What I'm trying to get at is the witness is a property of a solution to the problem, not a property of the problem itself. SAT and subset sum seem to have "natural" witnesses, but there still could be other witnesses and some problems may have witnesses that are less natural. Some witnesses can be restricted in size, and others can't.

With this in mind, there is still one major observation we can take in mind. If we limit witnesses to size $f(n)$ then there are only $2^{f(n)}$ possible witnesses so for small functions $f$ we can simply evaluate each possible witness to see if it works. In other words, by looking only at small enough witnesses, the problem is easy. This means if witnesses to inputs of size $n$ are guaranteed to be $log(n)$ in size, then the original problem can be solved in pseudo-polynomial time, and if the witnesses are guaranteed to be constant size then the original problem can be solved in polynomial time.