Some of the details are lost like maintenance details, advanced fault finding etc, but the core remains.
After a century of radio and electricity it would be rather silly to only search for other signs of life on the new system of FTL communication we just discovered. It still gives no certainty that the candidate alien discovered the same defacto standard for FTL comms or that theirs will be recognisable.
Any rational search should try and encompass both and probably still emphasise the analogue as any advanced civilisation would very likely go through that phase first, just like we are. We'd be likely to catch more possibilities that way.
Being able to send a message at a million times FTL would make it really easy to send messages back in time.
Actually that is exactly what physics says. Anything with whatever mass, as its relative velocity is accelerated, approaches a mass of infinity as its velocity draws nearer to c. Information, on the other hand, has no mass ...
In the 1890s, Lord Kelvin stated that physics was essentially complete. m'Lord was oh, so wrong.
Care to try again?
"Faster-than-light communication is, by Einstein's theory of relativity, equivalent to time travel. According to Einstein's theory of special relativity, what we measure as the speed of light in a vacuum (or near vacuum) is actually the fundamental physical constant c. This means that all inertial observers, regardless of their relative velocity, will always measure zero-mass particles such as photons traveling at c in a vacuum."
Such a relatively instant means of communication would become the defacto standard. Antiquated EM methods and equipment would be forgotten. Noone would be listening for it, just as very few today would recognize semaphore flags, smoke signals, or telegraph codes.
Today we'd not be eligible to join that club. And if we choose to remain certain that it doesn't exist, our determined ignorance would keep us locked out. If we conclude that we aren't special, that there -must- be life elsewhere, then the only logical thing to do is to keep our minds open and keep looking for clues.
We can't ever decide that we know enough. A century ago, the Milky Way was the whole universe. A few more observations changed that forever.