Does the phrase 'human race' allude to the idea of a relay?

No, those are two different words.

The race as in a ‘foot race’, is per the OED:

a. ONor. rás (Norw. and Sw. dial. raås), running, race, rush (of water), course, channel, row, series = OE. rǽs rese; of obscure etym. Orig. a northern word, coming into general use about the middle of the 16th c.

Whereas the other one, as in the ‘human race’, is again per the OED:

a. Fr. race, earlier also rasse (1512), a. Ital. razza = Sp. raza, Pg. raça, of obscure origin.

The first one has Scottish uses from the 1300s, while the other took a couple more centuries to first appear. Both came to prevalence in the same century, though.

Although both seem to lead to obscure origins, no one thinks it is the same singular origin for both.

Update

The RAE says that the Spanish raza mentioned by the OED as being of obscure origin actually derives from a putative Latin *radĭa, and that of course from the normal radĭus. That means the same thing as our race as in the human race. (Not to be confused with raíz from Latin radix meaning root.)

Just goes to show that one should always check more than one dictionary.