Your proposed rule would eliminate a section of most

teams offense - the big midfielder alley dodge that usually gets set up outside the box so that an offensive player can build up some speed. Granted as teams are moving more toward positionless lax on offense it just becomes any big, fast guy to do the alley dodge. There was an attempt about 12-15 years ago where offenses were time restricted on getting in and out of the box. I think there was a ten second clock once you touched it up in the box that you had to return to touching the box again within 10 seconds. There was also the 2 minute rule wherein a team that was up had to keep it in the box when they had it on offense. I believe they got rid of that rule when the shot clock was introduced but it never really impacted the game that much.

What it would likely do is push teams to play zone and become really compact in front of the goalie knowing a team would have reduced amount of field to work with in that last 20 seconds. Defenses might (there would likely be some analytics involved on whether this makes sense) then press out in those last 20 seconds knowing one bad step would result in a turnover. Offenses would likely wait it out knowing as a team presses out openings occur. So instead of creating earlier offense it would likely do the opposite.

The game is plenty fast as it is. Pushing it more can lead to more of a helter skelter approach. You would take the team aspects of the game out of it and move it more toward a professional product - think NCAA v. NBA. It move the game more to the PLL and predecessor MLL which I always found lame.

Further, several high level youth lax conferences (e.g., the elite division of the 7th and 8th grade HOCO lax conference) are going to have 7th and 8th graders conform to the NCAA shot clock rules next spring. So a shot clock will be coming to high school soon enough. Some intra-sectional lacrosse games like between private schools playing another private school in a different area (where most of the talent will end up in DI) have played with a shot clock.