I've become a bit of a tackle junkie. I fished exclusively with a fly rod, a 6 wt for many years, the result of a house fire that destroyed all my equipment, and at the time the only thing I replaced was the fly rod. Since then, I've added quite a bit of tackle to my arsenal.
I thoroughly enjoy fishing with a fly rod, but experienced enough situations where it was not the best choice at the time. It's admirable for anyone to wish to stick with one type of tackle, however, I feel that I missed some great fishing by doing so.
Just an example, several years ago, I was in Orlando, Fl & drove up to Dunnelleon to fish with a friend who had moved to that area. This was my second time on the Rainbow River, but first time fishing with this friend. I took my 6 wt along, but that day spinning gear & senko type plastic worms were the ticket for success as the bass were up under the over hanging trees, and the worm had to be skipped under the trees. The fly rod left me fishing the occasional open spot, and since we were also drifting with the current, I was making one cast for about every 15 or 20 my friend was making. He was catching plenty of fish, I was fishless! Even roll casting, I could not get the fly back far enough under those trees to where the bass were.
Fortunately, my buddy had extra spinning gear & I switched & had a great day of fishing. He still out fished me, but that was fine with me!
Such experiences have taught me that having the right tool when needed, may be the difference between a good day of catching, or a day of casting practice!
Of course, any day fishing with what you have available is still better than not fishing at all!
Now, in that scenario, had I been fishing alone with only the fly rod, I would certainly have tried to figure something out that worked!
IMO, fly fishing has improved my fishing with other tackle, but I also feel that the diversity of using other tackle has in turn, improved my fishing with a fly rod. I've been in situation that I needed to figure out what would work when using bait casting or spinning gear, and once I did, at other times I've been able to relate using a fly rod in similar situations. Doesn't always work that way, as others have said, because of the limitations of each type of tackle, but enough times that I feel it's made me a better angler generally.