Hm - even without the US we would still want to be allied with the others, for us it is particularly important to cooperate with the neighbouring nations, France, Germany, UK and the Netherlands, Denmark.
Belgium is very small, modern artillery shoots straight across it, our F16 pilots can hardly complete their post take-off checks before they reach the border, only way to operate a military here is in some form of multinational alliance.
NATO's primary function is ofcourse to protect the trans-Atlantic trade routes and the Channel.
In more general terms - you either compete or cooperate, it is not practical to compete with all, so you ally with some to compete with others, in this manner you shape and control your surroundings which is ultimately beneficial to the individual.
Alliances have been formed since the beginning of history, it's a force multiplier, cowboys join a posse, criminals join a gang, nations join an alliance