I'm sure teams would want to do this but the players would also be able to hit the ball which doesn't always come with a speedster, if a player does get on and steals 2nd and 3rd that's where it stops so you need someone to get an RBI. They could sacrifice him in but that means playing small ball really. Plus professional catchers especially the good ones have amazing arms and even the fastest guys get caught. Also lots less homeruns which accounts for probably a third of RBIs
A team of all-speed players does no good if they can't get on base. There are plenty of fast guys who don't hit for average or walk - they're generally not famous because they're not very useful.
In addition, power will score you more runs than speed.
Speed kills, you are completely right. Its not easy to get all speed players though. I cant name a speedy catcher, and usually first basemen are not that speedy. There are, of course, exceptions.
It would be cheaper to employ this strategy. Speed seems to me to be the most underrated aspect of the game. If you have a team with nothing but speed demons you increase your chance of getting hits via infield hits and you get more stolen bases. Has this strategy ever been tried to a large extent? I know the Angels a few years ago had a really speedy team. Any good examples of teams in history or any god reasons why this wouldn't work?