It sounds crazy, but it would be awesome. If the US Open Cup was open to the top 5 tiers in the US + NCAA + NAIA + HS state champs, it would be ridiculous. The travel would be the major factor, and if the NCAA would allow anything to happen for this, qualifying would have to happen in the spring, but could it be done. MLS teams would enter in the Round of 64 or 128 (only two more games to play), and there would be multiple qualifying rounds. There is a very similar tournament in Japan called the Emperor's Cup, which is open to all professional and the best collegiate and high school teams (it has thousands of teams in qualifying). It had a remarkable story in the early 2000s as a high school team got all the way to the Round of 32 and took the best professional team in the country to penalties before losing. If something like that happened in the US, it would be sick and it would improve the stature (IMO) of the Open Cup.
It would be fun...a logistical nightmare...but fun. They haven't quite figured out how to run the tournament with four tiers yet, so its asking a lot.