I don't see why someone who makes no claims about their previous salary is riskier than someone who proactively provides a pay stub. If Bob is inclined to lie in general, either he could be giving a true pay stub to engender confidence and hide some other prior/contemporaneous/subsequent lie, or he could have doctored the pay stub.
Other things being equal, neither Bob nor Fred is more likely to be a liar. Should I want to verify salary history, I'd seek permission to verify directly with the former employer (a routine request for any HR department assuming employee consent). If anything, it might feel suspicious that Bob was proactively trying to give me the easier form of proof to render fraudulent; but that alone wouldn't seem concerning in the absence of other signals, I expect.
Note that before the new law, the employer could legally have insisted that both Bob and Fred give their salaries or refuse to consider them. If Fred was severely underpaid and Bob was not, the employer would almost certainly have given Fred a lower offer than Bob. Since persistent underpayment also correlates with a more precarious personal financial situation, and since any other employer could legally have insisted on the same policy, Fred would probably have ended up accepting a comparatively low-ball offer for lack of a better option, thus staying underpaid.
The law still helps.