For me, in addition to what people have already stated multiple times here about this having been the TOS all along and Yoshi having made multiple verbal/written "warnings" to the general public over the years about it, before now, I think there's also some element at this point of the "suffering from success" meme at play.
FFXIV, at this point, has more eyes on it than it ever has in the past. Therefore, it's important that they maintain the image of integrity in the game if they want people to continue to have faith in them as a dev team and their product. If there are allegations of unfairness/cheating in a fairly publicized/currently popular piece of game content, it's in their best interest to do what they can to publicly remedy that situation ASAP to show they will not tolerate that kind of behavior and people can trust them to enforce their own rules. People reported someone flagrantly violating the TOS in a public sphere (i.e., streaming), so SE's options then were to do nothing and look like hypocrites and undermine public trust, or to hand down some kind of warning/punishment/whatever you want to call it for breaking their rules to the point where you've "riled up the commonfolk" enough that they're bothering to report you.
Do I get the impression from Yoshi's most recent message that they're going to necessarily be actively going around searching for such people at this point? Still no.
Do I get the impression that they absolutely could start doing that at any time that they wished because it is indeed a violation of the TOS that every player must agree to in order to use the game? Absolutely.
Do I get the impression that they are closer to taking some kind of widespread hardline countermeasure against 3rd party addons now than they ever were in the past, for a whole variety of reasons, many of which have been discussed in this thread? Again, yes.
To the point of whether a 10-day ban was justified vs. some kind of lighter warning, in my mind, absolutely yes. If Yoshi literally just put up that long message about "don't do this" and you're still very publicly doing exactly what he said not to do - no matter how you think the rules were or weren't enforced in the past - you're basically asking to get slapped. A "don't do this again" verbal warning with no actual consequence other than a strike on your account (which, again, we don't know if this person may have already had other previous strikes on their account) will do nothing to show you mean business. If you roll into work 10 minutes late on a consistent basis and your boss tells you not to do it again, does that usually mean you're NEVER going to do it again if there's no "teeth" behind the request? If your boss tells you not to do it again or there will be consequences, on the other hand, are you willing to do it again and find out what those consequences are? If the consequences are "up to and including suspension/termination," if you're in an area that has employment-at-will, is that something you want to test out to see if they're serious? Seems a bit like Russian roulette, to me... this feels the same as that, though arguably lower stakes than a job situation (though, maybe not even, if you're a streamer).
TL;DR: My opinion is that the ban was justified after many previous public warnings not to do exactly what the streamer was doing, and to show publicly that they enforce their own rules in this current era of unprecedented popularity for XIV.