Yes.

You're assuming more than what's necessary.

For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):

 - (a) bounded orbits
 - (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")
 - (c) axial (preserves an axis, on which some element acts loxodromically)
 - (d) focal (fixes a unique point at infinity, existence of a loxodromic element)
 - (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.

(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $\ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.

The case of actions on trees is a useful motivating baby case illustrating the above "classification"; all cases can actually occur.