When two metals at different temperatures are allowed to attain equilibrium in vacuum through radiation, theoretically they should attain it faster as compared to when the same metals at different temperatures are in vacuum connected by a metal rod. This reasoning is obvious because the same amount of heat is being transferred. In the first case all the heat travel at speed of light but in the second case some of heat travel at c while the rest travel very very slowly (through metal rod). Even though in the second case the two process occur simultaneously it should take more time because compared to speed of light conduction is an extremely slow process. So we cannot say it with certainty that two modes of heat transfer are better than one Or can we?
Technically, your metals do radiate thermal energy at the speed of light (in a vacuum), but that doesn't mean that they transfer thermal energy rapidly.
If you shipped something by a Ferrari (compared to a truck), you'll move it faster with a Ferrari, but you'll move a lot more with the truck.
Radiation is actually a relatively slow way for objects to lose heat and is only significant for very hot objects.
The rate of transfer of heat for objects in a vacuum is
Q ∝ εδT⁴*
where ε is the emissivity (which is very small for metals because they are shiny (0.04 for copper)), δ is a very small constant (5.670367(13)×10−8 W⋅m−2⋅K−4) and T is the absolute temperature. You can see that the Q value for heat flux will be very small.
The absorption of heat is, for the same reason, very slow for metals.
Now consider conduction:
Q ∝ k∆T*
where k is the thermal conductivity (401 W⋅m−1⋅K−1 for copper) and ∆T is the difference in temperature.
The two metals will almost certainly achieve equilibrium much faster with a metal rod between them not only because conduction is much faster and much more effective but also because the metal rod does not hinder radiation significantly.
*Surface area and distance left out because there's not enough info and because they doesn't affect the conclusions.