I'm not sure what world you live in where golf players are celebrities but tennis players get no respect. The reality is that outside of football and basketball, people do not care about high school sports. Elite tennis high schools are normally wealthy and filled with overachievers who really won't have time to follow smaller sports between schoolwork and the sport which they play.
Your entire premise is that if high school tennis teams were good and thus cut themselves down to six people, the level of tennis would draw people to the match. This is demonstrably untrue. I've seen duals between truly elite tennis teams draw no fans. I've seen matches between blue chips with nobody to watch. And believe me, when my tennis team was very good my Junior year (several years ago now... we went 19-1 in duals), I didn't get any more respect than my Senior year when most of our top players graduated and we were slighly over .500.
Most elite junior players aren't going to care about high school because it isn't going to get them recruited. When they do play, it is about enjoying themselves and hanging out with friends.
Thus, the solution in high school tennis is the exact opposite of what you are proposing. You want players to HAVE FUN. This means:
1. Do not have too many players per court. Most people play tennis because they enjoy tennis. By having no cuts, you are depriving people of the chance to play tennis and thus enjoy themselves if there are more than two people per court.
2. Invite more people to Varsity, especially underclassmen players. As a young tennis player, there is nothing more fun than being the hero of a consequential match. Obviously you don't want to have players who re picking up a racquet for the first time playing Varsity, but in most cases that means more than six people.
3. Emphasize competitiveness between other schools, but not cutthroat attitudes within a team. People want to have fun and celebrate wins together, not pray that their teammates lose so you can move up a spot. If you try to build a team who believes in winning duals, the effort will show up. Also, kids will fight harder for people they want to impress than people they don't care about.
If you make these changes, you'd have more fun and a stronger connection between a team. That is what draws the elite players to high school tennis, but it also would encourage younger players to stick with tennis. There would be some casualties (ie the people outside the top 30 or so players), but realistically, they aren't going to play varsity anyways whether there are six people on a team or twelve. Outside of forcing college coaches to only recruit from high school tennis (which is impractical for about 1,000 reasons), this is the best high school tennis can do.