There certainly were some intellectuals and politicians who argued for NATO's disbandment, but the institution carried on for two mutually reinforcing reasons.

Firstly, NATO continued to act as a force for collective security. Although from the vantage point of 2016, the causes and consequences of dissolution and collapse of the USSR are clear, they were less apparent in July 1991 (when the Warsaw Pact formally dissolved). The breakup of Yugoslavia added more to these fears as it demonstrated that the transition away from communism might not always be peaceful and could spill over into other nations. Many Western European and American policymakers felt that keeping NATO intact was the best way to shepherd Europe into a more peaceful order. A very influential speech given at the Berlin Press Club in 1989 by US Secretary of State James Baker highlighted that NATO's continued role in the post-1989 world. "A new architecture," he asserted,"must have a place for old foundations and structures that remain valuable- like NATO- while recognizing that they can also serve new collective purposes." For example, part of NATO's repurposing in this period was that it provided the apparatus to ensure the efficacy of the Open Skies treaty to guarantee nuclear disarmament. Furthermore, the breakup of Yugoslavia highlighted a new potential role for NATO: peacekeeping. NATO-led peacekeeping had greater flexibility than UN missions as it was far easier to create concord among 16 nations than the global community of the UN.

The second, and more prosaic, rationale for keeping NATO is bureaucratic inertia. Rather than being a part of a coherent geopolitical strategy, the continued existence of NATO reflected the embedded governmental networks and mentalities of its leadership and elites. It's important to realize that unlike the Warsaw Pact states, NATO governments really came out of the Cold War politically unchanged. Even the newly united Germany was fundamentally structurally identical to the Federal Republic, whose legal and state apparatuses displaced those of the GDR. Creating a new defense structure would have been costly and time consuming. Disbanding NATO would have also had a negative effect upon its members' own militaries. NATO created a force and base structure that spanned the whole half of the continent. If, say, the British Army of the Rhine were disestablish itself, the UK would have to find a place for these troops in the British Isles, constructing new bases and incurring significant costs. This is one of the major reasons why the US maintains such large bases in Germany; it is much easier and cost-effective to have units and heavy equipment proximate to trouble spots in Southwest Asia than having them return to the continental US. NATO airbases in Turkey were a vital component for the Coalition's air offensive in the Gulf War. Both Turkish and German bases' utility in this conflict proved the importance of keeping these military ties intact. In short, NATO continued to undergird American military strength. Similarly, other NATO countries would have found their military forces decidedly unbalanced without American assistance (heavy airlift and AWACs for example are only affordable to larger NATO states). Given this context, the repurposing of NATO as exemplified by Baker's speech was a welcome return to familiar territory.

Sources

Baker, James Addison. A New Europe, a New Atlanticism: architecture for a new era. US Department of State, Bureau of Public Affairs, Office of Public Communication, Editorial Division, 1989.

Hirabayashi, Donna M. To Be or Not To Be: The Future of NATO. National War College. Washington DC 1991.

Vincent, Richard. "Nato‐where next?." The RUSI Journal 138, no. 6 (1993): 8-12.