Mechanism to balance all game stages
I was thinking about dealing some tiles on the board in the beginning at random and limiting the total numbers of tiles available


You need to try it and see. It's the most natural solution, and that's what playtesting is for. If it turns out not to be fun, you can always bail midgame.

Other suggestions:


Are all the symbols the same rarity? If some are associated with more "basic" or introductory actions, you could choose a few specific starter tiles that contain only those symbols and start the round with those in play.
You could include cards that can be played for a reduced effect without the appropriate matching symbols. This would encourage players to match the symbols, but give them something to do early on while the board develops.
As midgame passes, perhaps players move to replacing existing tiles rather than placing entirely new ones. This would keep the board from becoming too full of options while not cutting the round short.