Probably not always
As Oliver Wood explained, in at least one Quidditch match substitutes were brought in for exhausted players:
"A game of Quidditch only ends when the Snitch is caught, so it can go
on for ages—I think the record is three months, they had to keep
bringing on substitutes so the players could get some sleep."
—Harry Potter and the Sorcerer's Stone
It seems likely from this that professional Quidditch teams have proper backups.
Hogwarts is a different situation, but as indicated in the question, they at least have substitutes.
As to why the team was a player short for the match in question, the explanation is probably simple: they did not have sufficient notification to find a substitute, or all potential substitutes were busy.
It seems possible that they did find a substitute, but still lost, because the substitute was not as skilled as Harry. This could still be consistent with calling them "a player short," since that might refer to the situation before they could find a substitute.
That said, in the Hogwarts Quidditch games, they do not seem to have a regular squad of substitutes,and instead must find them as the need arises.
For example, when Katie Bell ended up in the hospital, Harry had to find a replacement:
Katie Bell was still in St. Mungo’s Hospital with no prospect of
leaving, which meant that the promising Gryffindor team Harry had been
training so carefully since September was one Chaser short. He kept
putting off replacing Katie in the hope that she would return, but
their opening match against Slytherin was looming, and he finally had
to accept that she would not be back in time to play.
—Harry Potter and the Half-Blood Prince
The unofficial replacement roster seems to be those people who performed well at the last tryouts.
One of the reasons why Harry would have preferred not to have to ask
Dean to play was that he knew Seamus would not like it. On the other
hand, he had to do what was best for the team, and Dean had outflown
Seamus at the tryouts.
—Harry Potter and the Half-Blood Prince