They do, but the alliances are unstable. The problem is in determining what you mean by compete. Militarily that's next to impossible. The advanced weaponry the US, Russia, and China can wield requires an advanced technological society to produce. Many smaller countries just don't have the infrastructure, and worse would be hard pressed to access the materials to make them. Even fpv drones, like the Ukrainians are using against Russia, require advanced computing technologies that take decades to develop internally. But unless the countries have all of the materials within their borders they'd have to trade for it and be at the whim of the superpowers who control most of the world's supply. The science is available for much of the tech with maybe a 20 year lag time, with more current science being highly controlled and classified.

Economically there is more leeway. If the allied countries had natural resources or cheap labor they could trade that with other nations, but some resources such as oil, lithium, and rare earth metals pose an serious and deadly risk for any government looking to leverage them on a global market. Dozens of countries across the world have been destabilized and overthrown by the CIA alone, nevermind the 20 or so countries the US has directly invaded and dismantled for favorable oil trading.

Realistically a small group of countries allied together is the same as the individual countries going it alone. You'd need a larger alliance that is continent scale to have any real power when dealing with the superpowers. But even that is no guarantee as the US/EU relationship demonstrates. Without military strength the alliance would have no hard power to wield on the world's stage. Any nascent alliance would have to overcome meddling from the CIA/FSB/MSS if they started to become successful and threaten superpower interests.

Realistically the only countries on Earth that are sovereign are the superpowers, the rest are subject to their demands.