Because while the East German government was technically sovereign, the Brezhnev doctrine meant that if the government was to do something to change from socialism to capitalism (such as removing border fortifications) it would intervene militarily. Not that they even had to worry about this, as the East German regime was not going to take down the wall unless made to.

East Germany was essentially a Soviet-run country during the Cold War. It had some autonomy to run its own affairs but had no autonomy whatsoever with regards to things that had international consequences — like whether there should be a Berlin Wall or not.

Reagan addressed Gorbachev in part because Gorbachev was making a name for himself as a reformer, towards a more humane form of socialism. The speech then is a direct challenge to this notion of Gorbachev:

We welcome change and openness; for we believe that freedom and security go together, that the advance of human liberty can only strengthen the cause of world peace. There is one sign the Soviets can make that would be unmistakable, that would advance dramatically the cause of freedom and peace. General Secretary Gorbachev, if you seek peace, if you seek prosperity for the Soviet Union and eastern Europe, if you seek liberalization, come here to this gate. Mr. Gorbachev, open this gate. Mr. Gorbachev, tear down this wall!

He is saying, if you're claiming to be a reformer, you should start here, at this wretched symbol of Communist oppression and Soviet control. Reagan's speeches abroad were often more directly calculated for a domestic (American) audience than an international one, it should be noted. Not all of the USA's allies even wanted Germany to reunify (because they saw a strong Germany as a potential threat).

It is worth noting that the head of the East German government at this time, Erich Honecker, was not eager to reform along Gorbachev's lines and would not have been eager to remove the wall at all.

As for what happened in East Germany towards the end, it is a long and complicated story. The basic question you seem to be asking is, did the Soviet Union allow it? Essentially, it did. It allowed Honecker to be replaced, and when the border opened up, it did not intervene. It participated in the procedures that allowed Germany to re-unify as a sovereign state. By this time the USSR was having its own problems and it was itself dissolved within a year after Germany re-unified.