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It should do exactly that if you configure it correctly afair. Can you share an MCVE where it does not work, or possibly a build scan? I'm not sure whether the build scan contains relevant information in that case, but chances are there is.

I looked at the number of store operations for both types of build cache. Remote build cache always has 0 stores, whereas local cache has non-0. Also in the timeline, for a task that has outcome = SUCCESS I see that build cache result has miss and store for local build cache, but no store for remote build cache.

As I said, do you have an MCVE or build scan that shows this? Maybe you configured the remote cache to only be pushed to on CI and you run it locally. Or you run it on CI but the check is wrong like checking environment variable

CI

while running on TeamCity that does not set that variable by default and you don't supply it. Or you didn't enable the push to the remote build cache at all, as the push is only default on for local and default off for remote cache like documented. ...

The configuration is pretty simple, I doubt that there is any misconfiguration. The build scan clearly shows that both caches are enabled and push to remote cache is also enabled.

Well, then I am out of ideas if you can neither provide an MCVE showing the behavior and / or the build scan.