TAI is the 'UTC without leap seconds'.

UT1 is an evolution of solar time, and wobbles when Earth does because it's using the Earth and Sun (and other bodies) as a clock.

UTC is TAI with leap seconds added when UT1 drifts too far away.

Anything that cares about really small timescales would have issues. UTC and TAI seconds, microseconds, and nanoseconds are always the same length. UTC has leap seconds to account for the change in rotation of the Earth. TAI just doesn't care and keeps chugging. UT1 instead varies the length of the second so each day contains the same number of seconds. The variance would be less than 12ppm (1mil/seconds in a day), but some applications like navigation would have issues.

EDIT: Most computers end up using a modified version of UTC where the leap second is smeared at the NTP level instead of explicitly experienced. So any computer that uses that could probably use UT1. But we couldn't use UT1 universally.

If you set your clock to UT1 then it will be “wrong” by up to 0.9 seconds fast or slow compared to UTC, and the length of your seconds will be “wrong” by up to about 1e-8 compared to SI seconds. (Before its recent speed-up, the earth took about 24x60x60+0.001 seconds to rotate.)

One of the key reasons for moving away from UT1 or UT2 as the standard broadcast timescales up to 1972 was that the time broadcasts were (and are still) also frequency broadcasts, so when they adjusted the length of the second to match the earth, they also had to retune their high-power transmitter so that the advertised number of cycles per second was correct. And when this retuning happened they also had to promulgate the new adjustment factor to all their users who needed actual SI or ephemeris time seconds instead of rubber UT1 seconds.

It would be quite bad.

See this page: https://www.ucolick.org/~sla/leapsecs/dutc.html

And scroll down to "Implementing the plan outlined at Torino", taking note of the "Notable event" column.

Edit: I think I misunderstood your question. This table shows the results of what would happen if we just stopped adding leap seconds.