Rhythm is maybe more important in the sense that a tune with bum notes can be better than a tune with the notes right and the rhythm all over the place. You can tap out a tune often on a drum or a table top, and it can be recognisable.

But that does not mean that all songs ought to be sung with a steady mechanical beat. Nor tunes either. That's especially true of course of things like slow airs and the equivalent in songs. I agree with Jon that in songs often what matters most is the narrative or lyrical structure, which may require a very irregular approach. Still rhythm, but free and flexible rhythm.

It's the same with spoken verse - for example, in a spoken monologue, the narrator will slow down, speed up, pause and look around...