Here's my two cents on Team Builder.
Some background on me, for a touch of context; I've been playing since 2013, just before the release of Thresh, and every season since season 3 I've been Silver. I could probably be higher, but I just don't want to invest the time in ranked - I'm stressed out enough without obsessing over my LP.
I don't know what Riot intended for Team Builder, but I can tell you what it was, at least to me; a (mostly) stress-free environment to pick up the mechanics of a new champ or role, get a guaranteed game with the most recently released champion or rework, test out a new build or leveling path, or just burn time for some cheap laughs with nonsense team comps. I, personally, never put much value in a Team Builder win; yeah, it's nice to win, but it was always kinda meaningless. Which was a good thing; Team Builder accomplished what bots wants to, but can't and never will, and that's providing players with a decent experience to do whatever they like while also honing their skills with something new, or brushing up on something old. If you wanted to play a champ you just picked up, you went to Team Builder. If you wanted to play the new champ and didn't mind a mirror match, you went to Team Builder. If you wanted to come up with some nutjob gimmick comp or even just field random champs in random positions, you went to Team Builder. Almost everyone understood what Team Builder was, what it was for, and why they should just chill the fuck out. If you wanted to try something new and didn't mind long queue times, Team Builder was your go-to. I've played blinds once since 2013, and it was terrible. It was the bad of Team Builder (not being able to ban anything) combined with the bad of drafts (everyone takes everything too seriously); I understand that new champ select is trying to accomplish what Team Builder was meant to, but it really shows Riot's mindset that they didn't grasp the things that made Team Builder worthwhile and, dare I say it, fun. Especially considering the alternative to new champ select and ranked is...well, blinds. Or one of the other maps, but hey. New champ select is fine, now that people understand how it works; it's easier to avoid snafus of accidentally banning a champ someone wanted, you can form a strategy around the types of champions your teammates are picking, and you can (usually) get a role you want without having to rely on the charity of strangers. As a replacement for old draft, it's absolutely marvelous - but, it's also absolutely no replacement for Team Builder. It probably never will be, in my opinion - people see "Draft" and put on their tryhard pants, even though it's just a normal game.
tl;dr - in my opinion, Riot did a great job updating queueing and champ select for normal games (though I'd argue that there are definitely some tweaks to be made), but made a huge misstep when they chose to cut out Team Builder entirely. If they ever hold a poll asking players whether they'd like Team Builder back, I'll be saying "yes" in a heartbeat - and telling them a lot of what's written here, if I get the chance.