Thanks for helping me to organize my thoughts. Yes, I do exercise other than cycling and recommend such.

It's all about recovery. If one wants to improve cycling performance as much as possible, then the principle of specificity applies. Anything that requires recovery should be cycling specific. The wall that one runs into is hormonal depletion. If one's glands get tired, one can't go hard anymore. Heart rate doesn't come up, power during hard efforts falls off. One grabs a lower cog on familiar hills. One is overtrained or at least overreached. So that's the limit and the knife edge one walks if one is trying to improve as much as possible in any aerobic sport. We train as hard as possible without going over that edge.

For instance I've used a training program which recommends cross training early in the base period. Perhaps that's in there to increase saddle time more slowly than aerobic base work, or perhaps it's there to help strengthen connective tissue early on, in either case to reduce the risk of injury. For a few years I rode the Stepmill at our gym when the program called for cross training. However it didn't help my cycling at all and I finally quit doing that. It increased my aerobic training load was all.

I have found that some forms of cross training are helpful:
Cycling-specific weight work in the gym definitely helps. Even though it adds training load from which I have to recover, it's worth it. However I do longer workouts in winter as part of base training and then greatly shorten them in duration and frequency when I start doing hard rides and intervals in the spring.

Walking/hiking in zone 1 or below as a recovery tactic definitely works. I can substitute hiking for a day off in any training plan with no penalty. I can go hiking in the mountains the day after doing an extreme ride in the mountains with no problem and the day after hiking I can do form work on the bike, which I would do anyway, no problem.

Walking could be substituted for hiking on a day off just fine, just not aerobic walking. Zone 1 or below.

However when folks talk about running, they're running for aerobic fitness and that creates a glandular training load from which one must recover, while not adding cycling specific fitness.

A couple further examples:
this time of year I'm done with my hard rides and looking forward to our annual 10-day backpack in the Cascades or Sierra. So now I'll switch gears and start training for that. I've almost stopped cycling. I'm increasing my gym time and hitting the muscles and ranges of motion that I see hiking and don't see so much on the bike. I'm starting to run a little, but carefully so as not to get injured. Running is hiking specific training.

I have a riding buddy, much stronger and younger than I, who decided to run his first marathon with his son this summer. He disappeared from our riding group. I haven't seen him in months. I haven't even seen him since his marathon as he's presumably recovering and trying to get back into good enough cycling shape to keep up with the group again. Specific training works.