I contend that yes they would.
Listen for the buzzer in this high-school game in Texas at the very last second as ball goes in. This goal got coverage all over the internet, including Yahoo! and USA Today/sports.
MLS could use the extra coverage that 'buzzer-beaters' bring. More publicity wouldn't hurt. The 'buzzer-beater' is even more exciting than an extra-time goal because the game is officially over. Take a look at the team streaming onto the field.
Even though there are plenty of extra-time goals in the sport and they are fun to see, I believe there would be more end of games buzzer-beaters with a countdown clock (rule #1).
It's an American ideal and players want to be heroes. Playing till the buzzer would provide more goals and give fans more drama and common sense in their sporting event.
We're lucky to see the different effects that subtle rules changes in soccer can have from the high-school game and college game because these changes are sometimes the only comparison we have versus FIFA's 'Laws of the Game.'
The professional leagues, MLS-NASL-USL, are governed by FIFA and can't deviate from what FIFA prescribes without permission from them. If a pro league was to deviate, it would be considered a rogue league and players playing in this kind of league would lose playing privileges for their respective National Teams.
While it's true that MLS experimented with having the clock count down early in its existence, referees still controlled when the halves would end (see link in above paragraph). There were no actual buzzers to beat.
There are a variety of rules that have a slight twist to them from high-school and college soccer. For example, substituting players in at different times so that they may come out and back into the game is permissible.
As March Madness approaches, the term 'buzzer-beater' will be heard often. Arguably, it is the item that has driven the success of the NCAA college basketball tournament more than anything else.
College basketball is the sport that got the phrase to be so well known. MLS should take a cue from college basketball and ask permission from FIFA for a trial period in order to have more of its own 'madness.'