It's been a few years since I've worked with this stuff but I'm under the impression that you can do this sort of replay only for a short amount of time. If so, is there a point?

If the receiver expects a key to have been revealed at a particular timestep, it won't accept a replayed message with that key after that, so you can't record and replay indefinitely.

EDIT: Unless you indeed meant to instantly replay - would the receiver accept the highest strength signal, ie. yours?

> […] (just the time is slightly off/late, but that doesn't help much without a reference to check against.)

The 'time sync' does not need to be done in the same absolutely sense (time_t is the same everywhere), but only in the relative sense:

    Various approaches exist for time synchronization [15,16,17,18].
    TESLA only requires the receiver to know an upper bound on the delay
    of its local clock with respect to the sender's clock, so a simple
    algorithm is sufficient. 

Knowing the (rough) broadcast delay from sender to receiver is sufficient.

nobody would send a military vehicle (manned or drone) without initializing a proper clock, the replays would be stale.