Operator should work with RollingUpdate strategy normally, but in this case it will not restart pods in optimal way (primary last) and on database upgrade it will not optimally upgrade pods cfg, mongod then mongos (if I’m not mistaking), but it should work and I don’t suggest to try it in production environment (first try in test environment and if it solves your issues then apply to production).
SmartUpdate strategy is preferred and we test mostly with it enabled.
Do you maybe have some passwords/secrets rotation going on (in general, but for the vault also)?
Also are your ssl secrets maybe rotated and that can cause operator to restart pods.
About your error message which is seen in the logs there is some thread here: Cache Reader No keys found for HMAC that is valid for time - #7 by joaorpsoares
So in general I think operator is having trouble connecting/issuing commands to your cluster because of this error so it needs to be investigated more why this error happens.