I don't believe you need to be a musician to have a valid opinion on musicians. Ranking music isn't like ranking a research paper -- you don't need credentials. Just ears.
About The Eagles, they've become an easy target for criticism partly because of their massive popularity. IMO a lot of it is plain old snobbery. There are valid marks against them: Unlike the other bands being discussed, they're not groundbreaking in any way. No virtuoso musician among them. They didn't evolve much over the years or have a variety of styles -- songs and sounds remain similar throughout their career.
However, The Eagles found a really good sonic groove and wrote a wealth of really good material. Is that enough to make them 'great'? Depends on your criteria, because everything I just said about The Eagles could also apply to The Rolling Stones.
I know plenty of people won't agree with the comparison; that's cool. Very few bands can boast a 'wealth of really good material', so it's not meant as a criticism of the Stones or Eagles. Yet the Stones are considered among the greatest bands ever while The Eagles are often sniffed at as disposable pop. Why? I think the difference comes down to two factors: 1) Timing. Singing about raw sex in the 1960s had traction. The Rolling Stones were the bad-boy flipside to The Beatles' fresh British faces. Girls wanted to marry The Beatles, but they wanted to sleep with The Stones. The Stones' songs aren't better than The Beatles and not even better than The Eagles, but with their accent and attitude, they're much cooler than both. 2) Mick Jagger. Without the lips, the swagger, and the sneering vocals, The Rolling Stones would've been associated with a bunch of groovy, catchy songs, but not as an iconic pillar of an era. In other words, they would've been the 60s version of The Eagles.
Last thing in my defence of The Eagles. Their vocal harmony is superb. Not Brian Wilson-level superb, but still, one of the warmest, fullest and tightest blends you'll hear. Think vocal groups and Crosby Stills & Nash comes to mind, yet the level of vocal craftsmanship in The Eagles is better. Honestly, I wouldn't place The Eagles in any "Greatest" category, but I firmly believe the criticisms against them are mostly unfounded.