If fiat exists, and remains legal tender, it would only make sense to spend bitcoins when a person has run out of fiat.
So people will be buying bitcoins because it's a good store of value.
And they will only make payments with their bitcoins when they run out of fiat which they were holding for their regular payments.
Is that what you mean? If not, please explain. If so, doesn't that mean that they will eventually keep more fiat to not have to spend btc?
Pretty much, although bitcoins can be converted to fiat relatively easily (or even just given in lieu) if there's a fiat shortfall. If a company prices a debt in terms of fiat, which you don't have in this scenario, you either pay them directly in bitcoins or convert it yourself to fiat first then pay them that if it they don't accept btc. You don't really need a store of fiat, except to maybe hedge against bitcoin volatility.
The reason behind this is because of the inflationary nature of fiat against the deflationary nature of btc. Any dollar you hold is constantly watered down over time due to the increase in the money supply. The rough calculation is that it halves in purchasing power every 7 years. There will be no more bitcoins once the cap has reached, and instead be slightly deflationary as coins are lost. Assuming the network remain resilient, the purchasing power of bitcoins should go up over time.