Just to be clear - when you say "I'm evaluating the need for such a dearth of nonbasic fixing in the wake of a ten-fetch base.", I read that as "the need for such a lack of nonbasic fixing"...which doesn't totally make sense to me, especially in the context of the rest of the paragraph. Do you mean dearth? From the context it seems like you think you're running TOO MUCH nonbasic fixing lands, and would like to trim it down, so I'm assuming you mean the opposite of dearth (let's go with plethora?)
Assuming an ideal fetch/dual starting place, I think how much additional fixing you want depends a lot on what you want to do with the deck. Having 9 essentially rainbow lands (10 with command tower, 11 with vista) is usually a solid starting place, but if you want to be able to cast multiple CCC spells on-curve then more fixing is obviously going to be better, or if you just really want to have the right mana at the right time. 11 lands out of 30-something is obviously quite a few, but it's not so many that you're anywhere near guaranteed to draw one.
Personally, I tend to like to run quite a few utility lands (not like, a crazy number, but usually ~5 depending on the deck), so having some extra fixing to cover for the usually colorless slots is usually valuable to me. I rarely have any snow-synergy in 3-color decks (as in, never) so that's not an issue, nor is any monocolor basic synergy like gauntlets or whatever. So the only REAL downside to having all nonbasics is usually:
1-can't use basic-only ramp
2-worried about nonbasic hate
the first one is usually easily ameliorated by running a couple, if you want to run some ramp spells.
The second one depends on a lot of factors. Blood moon, for example, isn't THAT hard to play around with fetches, because as long as you already have decent fixing in-hand, you can crack for basics to protect yourself against it, and a lot of decks only need 1-2 mana of each color to work, which could easily come from mana rocks even if you don't have basics. There are some hate cards, though, most notably
ruination that...well, really, if that resolves in the mid/late game for a lot of 3-color decks, you're probably just wrecked. I don't think it's all that reasonable to run a bunch of basics and give yourself worse fixing for every single game just in case someone plays ruination. If that's common in your meta, then maybe it is, but it hasn't been true in any of mine. More commonly, I'll look around before the match starts and decide if I think it's likely that someone would be playing that sort of thing (revising that opinion as people play lands, of course). And if they are, I'll try to hold up a counterspell or something. My phelddagrif deck, for example - sure, ruination wrecks it hard, but so does armageddon. I've already got to play as though armageddon could come down, so I may as well play as though ruination could too. I've got removal answers for blood moon or B2B if those come down, but ruination is just going to be a bastard and it is what it is. Penalty of playing 3-color.
I don't think it's reasonable for most 3-color decks to run 10+ basics purely as a precaution against nonbasic hate, unless that's a major part of their meta. At a certain point, you've gotta just say "yep, if someone is running a specific hate card and it resolves, then I'm screwed, and that's just the way it is." If it's turn 12 and it's the difference between getting 9/10 lands destroyed versus 7/10 lands destroyed...who cares, you're screwed either way.