People can disagree with the claims of course, but I don't think they are baseless.

On the Postgres side: https://planetscale.com/blog/benchmarking-postgres

On the Vitess side, I would point to our customers, who, on individual databases, have achieved pretty high QPS (millions), on large datasets (100s of TiBs), at a latency that is lower than what other DBaaS providers can offer: https://planetscale.com/case-studies/cash-app