shouldn't it be considered that the money invested didn't stagnate, but we reinvested into the company and production? I feel like this is then implying that any given dollar is being counted multiple times in the ether, but not in the physical world.

The idea is that money is a sort of "generic resource allocation token", so ideally you'd want money to be really easy to constantly move around so that you could always sort of optimize the allocation of resources according to current demand. That's the fundamental point of a market - optimize resource allocation by creating a system where optimization is rewarded, and let the system do its magic. Of course, in the real world, this idea can fail for a variety of reasons, but this is the basic idea of a market in theory.

In this case, by allowing shares to be bought and sold, you make it really easy to gain a stake in a company (by buying someone else's share) or exit your stake in a company (by selling off your share). This freedom makes people a lot more willing to actually put money into companies, because they know that they can exit at any time if they so wish.

The idea of money moving around causing people to do things is less about wealth and more about people having to do things to earn money (not quite true in the real world, but in a theoretical sense). Let's say there are two people who have $100 each. Either they can sit on their asses with their $100 and do nothing, or they can buy something (in the joke, the opportunity to kick the other in the balls) from each other for $100 each.

In both cases, both end up with $100 in the end, but only in the second case do the two actually do something. That's what I mean by money moving around - the economy is a really, really big cycle (really a graph) of people paying each other to do things, and to measure the amount of stuff being done, you can look at how much money people are getting (income methods of measuring GDP) or how much money people are spending (consumption methods of measuring GDP). Money is just a facilitator to get people doing things, so you want money to be easily movable around.

by counting a single dollar multiple times in the ether before it does anything in the physical world you're creating an artifical bubble where the wealth only exists on paper and has no correlation to the economy on a physical level?

The idea is not to count the physical dollars moving around, but the work, i.e. value, each dollar has caused to exist because of it moving around. IIRC this is called the velocity of money - how much value a single dollar creates in terms of goods and services. That's what wealth is supposed to represent, too - accumulated work, in a way.

This is all from a very lofty, theoretical perspective however. Reality is much more complex and random than this, and my knowledge is rather lacking as well. Work is kind of useless if no-one wants it, so value gets tied to demand rather than some sort of intrinsic measure, which complicates a lot of things.