Propane tank sizes peddled by various local providers
Propane is stored and shipped as a liquid under pressure. That makes it convenient to store and ship a usable amount without needing very high pressures.
The pressure in a propane tank is a function of temperature.
When you use propane, the liquified gas in the tank boils. As it boils, it takes the heat needed to vaporize the liquid (latent heat of vaporization, from chemistry class) from the air around the tank. In summertime it's not uncommon to see frost forming on a 20lb tank attached to a barbecue grill.
So, the pressure in the tank drops as the liquid in the tank cools, until it can get more heat from the air.
Thus, when you want to use a fireplace (it's cold out) that happens ot be the worst time to place a heavy draw on a small tank. The tank gets cold, the pressure drops until it cannot supply the pressure required, it gives up.
Your disused connections, if not leaking, have no effect whatever on this.
Your best bet if you don't like renting a tank is to purchase a tank, and the best bet for that is to have it (correctly and to code) buried, where it will be warmer than it would sitting out in the air during winter weather.
Otherwise, you need to have a large enough tank that the rate of gas use does not overcome the ability of that size tank to supply gas at a particular outdoor temperature. Even with a buried tank, the rate of gas use must not be too great for the size of the tank, or it will still self-chill and fail to provide adequate pressure.
You can also manifold multiple smaller tanks (so if you are really dedicated to 100 lb tanks, you could gang 4 or 5 of them together rather than have a 420 lb single tank, if your local supplier, who does not appear to be dishonest, is estimating correctly for your local conditions.)