Here we may suggest a middle road between the unlimited-play
qualifier and the single-play: the Monty Hall format.
In the Monty Hall format, you have unlimited plays in the
qualifier, with this proviso: only your last round counts. When you commit to
play a new qualifying round, you agree to discard and supersede your previous
qualifying rounds.
In other words, do you stick with the prize (score) you
have, or do you give it up in exchange for the prize (score) that is behind
Door No. 1, Door No. 2, or Door No. 3? (The Monty Hall problem.)
This creates a strategic gaming decision, even when it is a
free tournament.
Instead of "grinding" out a top qualifying round,
players would have to assess whether the score they have already posted
represents about the best they can do, or is it worth throwing
out that score in favour of the unknown score on a new round?
Scores will tend to coalesce around each player's best
"handicap" score, i.e. about the best score they can normally turn
in when playing well. It would tend to collapse extremely low scores. For example, a player who posts a 54 would probably stick with that instead of discarding it in favour of the chance of scoring a 51 or 50.
In order to enforce the Monty Hall system, once a player
decides to play another qualifying round, the player has to complete that round
or be disqualified. In this respect it is single-play: that is to say, an
unlimited single-play.