Eudaimonia is translated by many modern scholars now as human-flourishing instead of happiness, or the literal translation of good-spiritedness. Aristotle studied medicine with his father, and spent years studying the zoology & botany of Lesbos. I read a compelling account I can't currently locate, that made the case that Aristotle's view of eudaimonia had a lot to do with his biological study. He described the telos of an acorn, as to become an oak - and we can see from that how a human achieving their telos, their flourishing, is their eudaimonia.
Teleological ethics aren't necessarily towards fixed goals. Flourishing, for an oak or human, can be in different ways as the landscape and events dictate. But it involves intentionality, goal-directedness in the most general terms, that even an acorn can be said to have (Greeks were much more animist, with a less sharp line marking inanimate). But it is a state of accomplishment, of options, an open-ended goal - in fact, the state that is most open-ended.
We can relate this to degree of 'freeness' of will. To amplify choosing with your whole being, that is with freedom AND self-knowledge, versus acting compulsively or in conditioned unreasoned contradictory ways, is to manifest wisdom (discussed here Wisdom and John Vervaeke's awakening from the meaning crises?). This is the real human telos.
For Aristotle the teleology is to attain eudaimonia. To accept Aristotle's definition of eudaimonia, is I think to accept it's what we move towards. So yes eudaimonia ethics are teleological.
I guess you could have an ethical system where eudaimonia is not the only organising principle, and indeed it is a bit woolly. But then it would be a hybrid system, or an elaboration on defining eudaimonia. I'd suggest that in dealing with practical moral dilemmas, eudaimonia ethics are anyway not enough, to solve difficult edge-cases, and recover what we think of as moral behaviours.