Just to clarify, the high speed at which electrons generally move around in metal is called "Fermi velocity". Like you said, since it's random, on average it cancels out to 0. When applying an electric field, the electrons achieve a non-zero average velocity which is called the "drift velocity".
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I hadn't heard of the term "Fermi velocity", but if it is the classical velocity derived from the Fermi energy, that should only be a lower bound on the average speed of the electrons in the metal.
That said, I don't remember the orders of magnitude here, the relevant question being how cold "room temperature" is with respect to the Fermi level -- a quick Google suggests quite cold, in which case the Fermi velocity should be a very good approximation.
In which case, it's interesting to realise that the motion of the electron gas comes mainly from Pauli exclusion rather than thermal noise! It's not a result I would have expected.