As to the fridge point, fridge vs. room temp probably wouldn't make much difference in what I glean of the scenario from a quick glance. It'd help a bit - would cool things a bit faster,* but, eh ... I wouldn't expect that to be a KEY element.

That written, temperature may still have been quite important (i.e., reboot alone wouldn't help). Particularly with a device containing something like spinning platters (as well as potentially extra heat generation associated with certain failure modes), and the additional factor of laptop heat load trade-offs, ... simply having the laptop off for some minutes might have been essential. In fact, the writer notes that transfer stopped during the first attempt to retrieve the data (after 10 minutes in fridge), hence a second round of 20 minutes was carried out after which the remaining data was retrieved.

That 20 minutes may have had little to no role in success the second time. It may have primarily been the difference between having transferred other files prior to transferring that (last? not checking that detail) file vs. not.

* Newton said proportional to temperature difference, but that's a rather rough guide in general for this sort of scenario - even with just basic details missing here