Using an UUIDv4 as primary key is a trade-off: you use it when you need to generate unique keys in a distributed manner. Yes, these are not datetime ordered and yes, they take 128 bits of space. If you can't live with this, then sure, you need to consider alternatives. I wonder if "Avoid UUIDv4 Primary Keys" is a rule of thumb though.
> you use it when you need to generate unique keys in a distributed manner
Just to complement this with a point, but there isn't any mainstream database management system out there that is distributed on the sense that it requires UUIDs to generate its internal keys.
There exist some you can find on the internet, and some institutions have internal systems that behave this way. But as a near universal rule, the thing people know as a "database" isn't distributed on this sense, and if the column creation is done inside the database, you don't need them.
I do not understand why 128 bits is considered too big - you clearly can't have less, as on 64 bits the collision probability on real world workloads is just too high, for all but the smallest databases.
Auto-incrementing keys can work, but what happens when you run out of integers? Also, distributed dbs probably make this hard, and they can't generate a key on client.
There must be something in Postgres that wants to store the records in PK order, which while could be an okay default, I'm pretty sure you can this behavior, as this isn't great for write-heavy workloads.
If one needs timestamp ordering, then UUIDv7 is a good alternative.
But the author does not say timestamp ordering, he says ordering. I think he actually means and believes that there is some problem ordering UUIDv4.