What America - and France, and Britain - agreed to was that only German local defense forces would be stationed in the former DDR, and German local defense forces in the DDR would not be equipped with nuclear weapons:
[quote=Treaty on the Final Settlement with Respect to Germany, Article 5]ARTICLE 5
(1) Until the completion of the withdrawal of the Soviet armed forces for the territory of the present German Democratic Republic and of Berlin in accordance with Article 4 of the present Treaty, only German territorial defence units which are not integrated into the alliance structures to which German armed forces in the rest of German territory are assigned will be stationed in that territory as armed forces of the united Germany. During that period and subject to the provisions of paragraph 2 of this Article, armed forces of other states will not be stationed in that territory or carry out any other military activity there.
(2) For the duration of the presence of Soviet armed forces in the territory of the present German Democratic Republic and of Berlin, armed forces of the French Republic, the United Kingdom of Great Britain and Northern Ireland and the United States of America will, upon German request, remain stationed in Berlin by agreement to this effect between the Government of the united Germany and the Govenments of the states concerned. The number of troops and the amount of equipment of all non-German armed forces stationed in Berlin will not be greater than at the time of signature of the present Treaty. New categories of weapons will not be introduced there by non-German armed forces. The Government of the united Germany will conclude with the Governments of those states which have armed forces stationed in Berlin treaties with conditions which are fair taking account of the relations existing with the states concerned.
(3) Following the completion of the withdrawal of the Soviet armed forces from the territory of the present German Democratic Republic and of Berlin, units of German armed forces assigned to military alliance structures in the same way as those in the rest of German territory may also be stationed in that part of Germany, but without nuclear weapon carriers. This does not apply to conventional weapon systems which may have other capabilities in addition to conventional ones but which in that part of Germany are equipped for a conventional role and designated only for such. Foreign armed forces and nuclear weapons or their carriers will not be stationed in that part of Germany or deployed there.
It’s also worth noting that this treaty was signed with a country that no longer exists, and whose last leader agrees that it only blocked the DDR from having permanent NATO forces and/or nuclear weapons stationed within its former borders.
Nobody in 1990 foresaw the dissolution of the Soviet Union or the total collapse of the Warsaw Pact, so there was never any reason to mention - or even imagine - Poland, Hungary, Ukraine, etc, joining NATO.