Not if you use a low TTL. And no, I'd rather not have my traffic run or even terminate through every intelligence agency on the planet, even if that means accepting a few drops in the timeframe it takes to propagate.
... hence the lose-lose scenario. Having a low TTL just makes it your client's problem. If you want to be paranoid about the government, you can still use an L4 load balancer such as the one offered by GCP.
Run your infrastructure however you want. This setup would never fly for a real organization.