In modern CPUs registers are a high-level abstract concept (see register renaming), so writing to a register doesn't have any specific location to overwrite.

Recent Zenbleed vulnerability was an example of that — clearing of a register was setting a temporary boolean saying it's zeroed instead of writing actual zeros.

I don't think so. This vulnerability is leaking data from a load-buffer, not directly from registers. It affects data that is loaded in bulk or to vector registers.

However, general purpose registers are also loaded passing through this buffer during context switches.