Not really. I mean, it's true that we should distinguish between the 2 and everything that you said about the difference is also true. But unixtime doesn't really "exist" in a sense UTC and TAI do. It is rather an imperfect implementation of UTC, that chooses to ignore (or repeat) some seconds.
You can hear that unixtime is the number of seconds passed since X. But it isn't really true though. The number of seconds is number of seconds, it isn't defined by our standards, it just exists. And TAI is a fair representation of how many seconds on Earth actually passed (on average) since whatever.
UTC kinda does it as well by virtue of 1 second being equal to 1 TAI second, but it actually counts (counted until yesterday) the number of Earth's rotations. It's just every rotation (represented by 24H) sometimes consists of more than 86 400 seconds.
Unixtime on each individual device counts nothing. It imperfectly represents UTC timestamp received over NTP. Some timestamps are represented twice by the same value. Of course, you can just put your device offline and call it "unixtime" whatever number of seconds it counts since any moment, but you know it will drift away from any meaningful "real" time soon enough.
(Also, it's not even entirely fair to say, as you did, that it is unixtime that was chosen by all the software. Many programs store datetimes as strings. Usually, they still don't support "23:59:60" anyway, but that doesn't really make them unixtime. Unixtime is a timestamp encoding.)
So, that's basically what I'm talking about: you can just make unixtime an implementation of TAI (as opposed to UTC). You can build a new calendar format for it, introduce a new name (not UTC!) for it and see how well it does when all the world slowly drift from Earth's rotation to keep up with TAI. Maybe it actually is fine, I'm not a judge for it (because it really is complicated and I didn't decide yet if it's a good solution or not). But why the fuck would you destroy UTC for it?! It is a closest usable representation of UT1, which doesn't stop to exist! Leave it be!