If elite players prefer to play short draws in knockout competitions, why not skip them and go straight to tiebreaks? From the Chessbase article covering the Jerusalem GP:

After day one of the quarter-finals finished with disappointingly short draws on all boards, FIDE Press Officer Anastasiya Karlovich made a point of asking the players about the controversial decision to choose pragmatism over combativeness. Wesley So responded impassively, pointing out that &quot;one of us has to win&quot; in the end. This is simply a logical consequence of using the knock-out format in events with so much at stake. Or, as Maxime Vachier-Lagrave put it:
&quot;I understand the frustration from the point of view of the spectators, but for me it's about something else, it's about qualifying for the Candidates, so for this one I cannot be entertaining, at least not if it diminishes my chances.&quot;

Elite players are unwilling to take risks to give themselves the best chances to qualify for the Candidates, so they play short draws. Okay (I'd even say it's quite logical).
But if they're going to play short draws, why bother with actually playing out the short draw on the board? Just skip the classical games and go straight to Rapid. Or if elite players play short draws in Rapid as well, skip that too and go straight to Blitz.
Conceivably some elite players might want to fight it out in classical games, but it doesn't sound too hard to accommodate them. Ask them before the match if they prefer to draw the classical games, and if either of them say no, let the match go ahead. But if they say yes, just save everyone the disappointment (not to mention the $$$) and play Rapid/Blitz, which sounds like a much more sensible use of time. One could even use the time freed up to, e.g., play more Rapid tiebreakers, get the players to commentate, or even give public simuls.
Why haven't elite tournaments done this?
It is because there are tournament regulations, including the format, which are published before the tournament starts. You cannot really change that after a certain point, like players accepting invitations.. In addition, although certain players are amenable to draw because they believe they are better at faster time controls, by FIDE rules, they cannot technically agree to a draw before the game starts.

Also, they might sit down at the board, and play some moves to see if they actually like how the game has started, and actually press on if they like what the see.

I think that what you are asking, if they tried to do it, it might be more complex in practice, and might also have unforeseen consequences, since everyone should play be the same rules. You cannot really have some players playing it, and others opting out since everyone should play the same tournament. Nevertheless, they could change the pre-tournament regulations and try it to find out if it works.