Imo, not necessarily. Theoretically, yes, because you can, as you’ve already stated. But if your nature is to hold on to cash almost as if it were a commodity instead of currency, then wealth will not automatically lead to investing or gambling.
The opposite human nature, of course, is to gamble or invest regardless of how poor because of the “what choices do I have? what do I have to lose?” type of mindset.
The motivational speaker/celebrity Tim R.b..ns is expert at convincing unwealthy people to throw money at his “self-empowerment” programs using the same mantra, “no pain, no gain”, no investment, no profit.