Is there a way to apply the same transition to many clips at once?

yup. select all the clips in the timeline and drag the transition on top of any part of the selected area. You might want to duplicate the sequence before doing this in case of operator error.

Make Timeline the active window.

Press command and A keys to select all clips.

Press command and T keys to apply the default transition.

This will usually be a one second crossfade but you can change it to another duration or a completely different effect if you want.

Link: Changing the Default Transition in FCP

or just drag the desired transition effect from the effect tab in the browser. You can also duplicate and customize the specific effect. Doubleclick on the transition in the effect tab, which loads in the viewer, customize and drag the effect to the favorites folder in the effects tab.