it's interesting to think about why not. it probably comes down to noise and friction in transactions, so that 1e-9 of your company is not a meaningful/tangible quantity. you could imagine an economy in which you could indeed issue a billion shares at a dollar each and have them be worth buying individually.
The reason why it is not is that the assumptions you have to make about valuations in order for the one dollar investment to translate to the company actually being worth one billion do not hold. You have to assume that the investor is a rational actor, is capable of correctly evaluating a company's value, and is making the investment because they believe the company is worth one billion dollars.
There is no reason to believe any of these things. Lots of people who satisfy none of these requirements have a dollar to waste. They could be investing as a joke, they could be trying to pump up the value of the company for some other reason, or they could be incompetent at evaluating a companies worth. Just because a person is willing and able to throw a dollar away doesn't mean you can find people to throw a billion dollars away.
The problem is that there is no connection between someone giving me that dollar and the value of my company; just because they say that they are giving it to me because they value my company at one billion dollars doesn't make it so. Maybe they are doing it to be funny, or to prop up the value of my company. They are only sacrificing a dollar, so there is no real cost to them. It also has no bearing on or indication that anyone else would also value my company at that much.
How much do I need to actually have to be a billionaire?