We looked at using UUIDs last year for a project in MySQL since we had good use cases for handling ID generation outside of the data tier. We originally prototyped using type-4 UUIDs, but found the locality problem made that a non-starter (updating indexes seemed to get exponentially slower when tables went over 10m rows). But switching to type-1 UUIDs made that problem go away. Still, we eventually chose not to use UUIDs since 128 bits made our indexes too large. At the DB level, we represented them as BINARY(16), but even with the pain that caused, our indexes were still too large.
Perhaps it's just a MySQL thing (I didn't make that choice) but, from our testing, I'd be very wary of using 128-bit IDs and would probably choose something like Snowflake or other centralized ID generation service.
Given what we've seen from the browser vendors lately (hiding bits of the address bar, etc), I'd say there's a lot of momentum to disagree with that statement.
(FWIW I like seeing UUIDs in my URLs, because it tells me that the company on the other end has its shit together. That might be a bit programmer-specific though)
For comparison, here's an eager.io URL:
https://eager.io/app/ZYBle8qUhKFJ
And here's an equivalent with UUID in base-64:
https://eager.io/app/b8tRS7h4TJ2Vt43Dp85v2A
It's a rare application for which those differences matter. (NB: I don't know anything about eager.io... it might be important for them!)
[0] https://blog.twitter.com/2010/announcing-snowflake