Best way to design tournament database
I think that using the team ID is the right way to go.  Another level of abstraction for all of the finals rounds just adds unnecessary complexity for not much benefit other than pre-loading the matches table with data.

The data structure looks pretty solid to support this.  The quarter and semi finals would need to be added to the matches table once the initial match results are in.  If the matches are assigned randomly then this is a manual operation, however, if they are in a particular order...

   A
match 1 -----+
   B         A
          match 5 -----+
   C         C         |
match 2 -----+         |
   D                   A
                    match 7
   E                   F
match 3 -----+         |
   F         F         |
          match 6 -----+
   G         G
match 4 -----+
   H


...then this could possibly be done with a query.  Again, the complexity of the query may not be worth the effort depending on the number of teams