Should you use Let's Encrypt for internal hostnames?

Ingrained practices are the sort of thing that change one funeral at a time (see constant password rotation).

It's a reasonable mitigation for certain environments and does leak information that makes structuring attacks easier, but it's certainly not a hard wall of any sort. The main problem for most people is articulating the realistic threat models they are trying to address and because that rarely resolves well assuming the conversation is had at all, there is little rational pushback against "everything and the kitchen sink" approaches based on whatever blog the implementer last read.

Personally I tend to advocate assuming your attacker knows everything about you except specific protected secrets (keys, passphrases, unique physical objects) and working back from there, but that's a lot of effort for organizations where security is rarely anything but a headache for a subset of managers.

You'll see similar opinions about things like port-knocking puzzles and consumer ipv4 NAT, which provide almost zero security benefit but do greatly reduce the incidence of spurious noise in logs.