Timing attacks are only a subset of side channel attacks, though. One can also imagine thermal attacks -- the amount of power you consume leaks information about what you're doing. And if I share a processor with you, there's various ways I can imagine estimating your power usage. On a processor that has dynamic clocking, the clock speed I'm running at is an indicator of the operations you're doing. Even without dynamic clocking, the probability of an ECC error, for example, is likely to change with temperature.

Eliminating timing vulnerabilities is necessary to allow potentially-hostile workloads to share hardware, but it is not sufficient.