Why don't companies just issue an infinite number of infinitesimal shares?

I am still a bit new to investing in general, and I'd like to understand this. The more shares a company issue, the more people will be able to invest because each share will be cheap and affordable enough to everyone. Why not create infinite infinitesimal shares? I know there is this thing called "fractional shares", but why bother with them instead of doing this?

I can understand why a company wouldn't want to issue more shares after their IPO: each share would be worth less and shareholders wouldn't be happy about it. But I don't see the issue doing this in the IPO itself.

When you buy shares, specially fractional shares, you usually don't really care that much about the number of shares you invested, but rather, how much money you invested and how much you profit from their dividends. Why even bother with the concept of "number of shares" when they don't really matter at the end of the day? Wouldn't it make more sense to say "I own X% of the company" instead of "I own Y number of shares"? Why are we doing things this way then?

If a company wanted to sell more shares (even if that would mean making their shareholders unhappy), instead of issuing more shares the traditional way, they would just say "We are paying less dividends for each invested dollar in our company but you can invest more now to get paid the same amount of dividends you were used to be paid" instead of "We issued 100 shares, but now we are issuing another 100 so you will have to buy more shares to keep the same amount of dividends".