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

Firstly, NATO continued to act as a force for collective security. Although from the vantage point of 2014, 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 highlights 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.

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. 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.

In terms of international relations and national security, NATO was formed for 'collective defence', but the Post-Cold War has made NATO continue for the reason of 'collective security' because the threat is gone (like you said) but NATO still exists because the threat could return in the form of a change in global interests (E.G. drought, and water availability).

Collective Defence is the concept where states ally themselves with other like-minded states against a perceived common threat or enemy.

Collective security is the concept of states forming agreements with not-necessarily like-minded states to enhance regional security.

I would also like to direct you to the works done by Sir Halford Mackinder, who influenced geopolitical thinking in the early 20th century. He created the 'Geopolitical Pivot of History' or also known as the 'Heartland theory'. This theory was developed around 1919, and a main quote of that is ''Who rules East Europe commands the Heartland; who rules the Heartland commands the world-Island; who rules the World-Island controls the world''. Mackinder saw that the land that was the heartland (Eastern Europe) controlled the entire world. This theory is still relevant today because many powers still see the influence of Eastern Europe. Today, this collective-security approach of NATO attempts to ensure that the buffer states do not return to the power of Russia (this theory can also attempt to explain why the Western world cares about Ukraine's sovereignty).

Hope this can answer your question and I'll be happy to answer any further questions you may have.