I would say I'm a fairly experienced hiker, and would love to begin climbing some more challenging mountains i.e. mountains that require actual climbing, rather than just hiking. [...] are there any benefits to learning how to climb outdoors, when the end goal is outdoor climbing?
The answer to your question depends somewhat on what type of outdoor climbing you have in mind: sport? trad? alpine? Since you describe climbing as an extension of hiking, then you are most likely to be interested in trad or alpine. In this answer, I'll focus on trad climbing.
If your goal is trad climing, then there is basically almost no overlap whatsoever between the skills you learn in gym climbing and the skills you need for trad climbing. They're almost unrelated sports.
Trad climbing involves learning belaying, routefinding, knots, placing gear, evaluating rock quality, rope handling, building anchors, and possibly multipitch climbing. Gym climbing involves none of these, except for belaying plus an extremely limited amount of knots and rope handling.
Gym climbing can help with building certain types of strength, endurance, and technique. However, as a beginning trad climber you would never get on a route that would require this kind of physical ability. Also, these physical abilities gained from indoor climbing are basically 100% about face climbing. Outdoor climbing involves slab, crack, and face, with crack being by far the predominant type.
Anyway, indoor climbing is fun, so who cares whether it prepares you for outdoor climbing?