I don’t think that most adults are opposed to scouts finishing their Eagle at any particular age. Rather, the objection I generally see/hear is to what @WilliamC mentioned: scouts who are artificially accelerated through the program, rather than those who accelerated themselves, as it were. Some of us were just fine as Eagles at 13. Other scouts I know who were significantly older wrapping up the requirements were “helped along” by a non-trivial amount of leader/parental involvement/pushing, and weren’t as ready/capable as they could have been if they had been more self-motivated. It’s unfortunate, but adult involvement it is a real concern/event.
I for sure would rather have a scout who gives back – whether or not he or she is an Eagle yet – than somebody of any age who does the equivalent of what we refer to as “sash and dash” in the OA. Whether that time is 5 weeks, 5 months, or 5 years, I think that the example of contributing to the unit is more important than the duration of that contribution. Would I rather see more time? Sure! However, what I least want to see is a scout who hits Eagle (or whatever their goal) and vanishes. While there may be many reasons it happens, often times what gets communicated (from talking to youth in scoutmaster conferences) is the appearance that the scouts are only in it for their own achievements, and then they’re gone.
I believe that Scouting is a continuum of experiences, not just a roadmap to a destination. We have had scouts who hadn’t yet reached First Class who were experts in navigation or knots, and ended up instructors for those skill sets. I’ve seen some “senior” scouts who regrettably can’t navigate their way out of a paper bag without instructions, either because they were “passed” without learning the skill, or they haven’t had enough practice to retain it.
Coming back to the OP’s question, I agree that encouraging the scouts to take up new challenges (e.g. officer or committee roles in OA chapter/lodge, or advanced training like NYLT and NAYLE) is a great way to keep them excited about scouting.
I would suggest also looking for ways to keep them involved at the unit level, by engaging them as “instructors”, JASMs or other skill-related positions where they can continue to contribute to the unit, and keep the goal of Eagle (not just earning the rank but modeling the behavior) in the minds of the other scouts. Adult association should not be the only way that servant leadership should be getting modeled.
I would also look at encouraging them to be actively involved in planning the troop and patrol programs, even if they’re not specifically in a leadership role. Having achieved the Eagle rank, they have likely been on a lot of hikes and campouts. Maybe they can help develop/enhance the program for teaching younger scouts these skills, or plan higher-skill-level events. I intentionally avoided the term High Adventure, because in some cases there are specific age limits that come into play with HA activities.
Most importantly, I would ask them what would motivate them to stay involved. It can be a great start to ask a bunch of “old folks in tan shirts” what they/we think, but ultimately the question is “What is the program that those scouts will want to participate in and lead?” That’s closely followed by “And what do we need to do to facilitate that program?”