This reverts #98150 because we think it introduced a problem that was detected
via the `replicate/wide` roachtest[^1]. It seems that for reasons still unknown
we do rely on the periodic gossip trigger on liveness lease extensions.

[^1]: https://github.com/cockroachdb/cockroach/issues/99268#issuecomment-1484968745

Touches #99268.
Touches #97966.
Closes #99268.
Touches #98945.

Epic: none
Release note: None