Here's why he's a really big deal as a photographer:
1. He, more than anyone else, made it acceptable to shoot color as art. Before, the assumption was that if you were shooting art, it had to be B&W as a photographer. Now we're used to film or digital that allows us to saturate or even over-saturate colors so Eggleston's work looks a bit "ho-hum." But in his time, he shocked a lot of people with the realization that color works could indeed be artistic.
Howzzat? People have been shooting colour film for art since the second day they sold it!
2. Eggleston is the answer for all the newbies who say "there is nothing to shoot where I live" or "I live someplace boring--no good shooting here." Eggleston made the mundane his subject. Rather than travel someplace exotic, he sought to find exotic (usually subtly) through the everyday commonplace world we all lived in.
Fair enough, but I would dispute that he is a good answer. I've looked at as many of his images as I can find since you posted the thread, and I would guess that fully 75% of his work is, IMO, simply bad photography. He does capture some interesting scenes, no disagreement there, but in a snap-shotish manner, with no concern for overall composition, cropping or exposure.
3. Eggleston exemplified the stereotype of America at that time. Rather than feeling like you had to be French (or foreign) or go to Paris or London to shoot good work, his works tended to be subjected that were strongly associated with America...gas stations, parking lots, diners.
Was there really such a sense at that time? As for the association, having traveled all over the world, I've seen gas stations and parking lots on every continent and in every city I've visited. I don't see anything especially 'American' about them.
There's no doubt he's done well for himself, far, far better than I ever will, and for that, I applaud him, but referring him to him as a 'great' or 'influential' photographer? IMO, that simply cheapens those adjectives.