pgcat VS postgres-operator

Not natively, no. You’d need to front it with something like PgCat [0].

[0]: https://github.com/postgresml/pgcat

For #1 I've been keeping a keen eye on pgcat [1], in particular the https://github.com/postgresml/pgcat/issues/303 which

As for the Operators, I've been using Crunchy PGO, which is very high quality, and one of the most widely used. You can install it via Helm, or via OLM from OperatorHub. There are other good ones as well, but none that I have experience with. The only issue I've run into so far is I've had to disable TLS on the database cluster, as Prowlarr refused to connect with it for some reason (Radarr was fine). I still need to open an issue with the Prowlarr team about that, but I might switch to a service mesh for TLS anyway.

The Crunchy operator, seemingly like most (if not all) of the other Postgres operators (Zalando, KubeDB, and StackGres, etc.), is essentially a wrapper for Patroni. IMO if someone wanted a Patroni cluster, they would just build one. The point of an operator is to manage the cluster resources and node relationships, so why not have it take the role Patroni is filling here? It's already reaching into the nodes, obtaining status, managing the routing, etc., so why add the extra layer?

On the topic of Postgres, you should look into an operator or Helm chart that can setup common things (like replication and auto-failover), such as Crunchy's Postgres operator, or consider using a "cloud-native" distributed database like CockroachDB (disclaimer: I am a Cockroach Labs employee) which has its own operator as well. Another word of warning, running stateful services, particularly mission critical databases, can require a lot of maintenance work (it's my full-time job), so unless this is for a hobby project, I would highly recommend you look into using a managed database offerring. Every major cloud provider and most database companies have one.

For postgres you can also have a look at PGO or bitnami helm chart

Let's say a new Kubernetes version comes out in April. In November, as everything works perfectly well, you decide to install a Postgres operator on it. Bummer, it doesn't work. It's not a huge issue, you just wait until the bug is resolved (already done[0]), but it's just one of these tiny things that I don't get when running Postrges natively. And I'm saying this as a big fan of Crunchy Data running some production loads on it without a failure for quite some time now.

[0] https://github.com/CrunchyData/postgres-operator/issues/3476

There is one particular client that have a somewhat big database 40-120gb (it change size over the year), and for that we used CrunchyData Postgres operator ( https://access.crunchydata.com/documentation/postgres-operator/v5/ ) we have no commercial relation with them, but oboi let me tell you the god send that thing is, this database in specific process massive data and it is distributed between several nodes in a read-write and read-only set, and let me tell you, it is amazing how easy it is to move things around, take backups, increase the capacity and a bunch of other goodies that operator bring. Give it a try.