it depends a bit on where you are in the game, but on the whole i'd say it's too extreme a stance for a few reasons. first off, if you sell everything that doesn't have a speed stat on it, you run the risk of finding yourself short on gear when you need it.
second, while speed is certainly important, it is far from the only thing you need to be cncerned with. if all you do is build your arena team up faster without concern for anything else, you will wind up with a completely ineffective team who can't land their debuffs and has terrible damage output. they'll do absolutely nothing, but they'll do it very quickly. even in a typical speed nuke team, the pinnacle of the speed is king meta, you'll have one champion who is at least as concerned with accuracy as they are with speed and another who wants to hit the minimum speed needed to avoid being cut in on and then focus entirely on crit stats.
third, i think we can agree that speed is king in the arena, so obviously speed is important when building a champion–if you're planning to use them in arena. i have 14 champions that see regular use between classic and tag arena and a couple others that get subbed in under specific circumstances. the rest of my champions are used solely for pve content and while speed is certainly useful there, it's not the end all be all that it is for pve. there are certainly champions i use there that i built for speed, but just as many where i'm far more concerned with accuracy or crit rate or hp.