The Fermi paradox and the gladiator ideas do not seem equivalent - unless someone erases the coliseum walls, or lies about previous gladiators.

For the Fermi paradox, either there's a late great filter, or not. In your setup, the evidence is strong for there being one. Then, assuming EDT or similar, we plan to spam info to the galaxy. If our plan succeeds, then, to the extent that we believe that other civilizations would follow the same plan, this pushes us away from the late great filter (actually there is a second hope - we could assume that we are different from all other civilizations - we spammed info to the galaxy, after all. Then if we act strongly conditional on this fact, we'll be exploring new approaches, untried by previous civs, putting us out of their reference class).

But for the gladiator, we can see that there are no names on the walls, and we know there were previous gladiators. Their failure is a fact for us, and sneakily getting our own name inscribed changes this not at all (it just tells we were atypical in this one regard, and unless we think we're atypical in relevant 7th victory regards, this doesn't help us). But if there was a possibility that names got erased, or that there were no or few previous gladiators - then our decision pushes probability in those directions, upping our chance of survival.

Anyway, cheers for the idea "should we set up powerful long-lived solar or nuclear powered automated radio transmitters in the desert and in space that stay silent so long as they receive a yearly signal from us, but then if they fail to get the no-go signal because our civilization has fallen, continuously transmit our dead voice to the stars".