I remember the explanation (mentioned in the SNW episode, too, IIRC) was that they needed Subspace Relays not to boost the strength of the signal so it'd go further, but to somehow make it go faster.
I think so too. SNW gives us some pieces to the puzzle. In LUWSCR, Spock tells the First Servant that subspace radio signals in long-range communications travel at roughly 52,000c. The First Servant then comments on that subspace relays are needed to relay the signals before they degrade but the distance isn't specified other than being part of long-range communications. In "The Serene Squall", the Enterprise is in a sector that is outside of subspace relays to provide "rapid communication". A message would take 2 days to reach Starfleet so I assume that the subspace message would not degrade after 2 days of travel (~280 LY) before reaching the subspace relay to then rapidly travel to Starfleet.
SNW appears to be taking the TNG episode "Where No One Has Gone Before"'s calculated speed of the subspace message back home of 52,000c. Interestingly, the TNG episode doesn't point out any issues with signal degradation at 2.7 million LY with no relays in-between but maybe TNG has better subspace tech?
The "rapid communication" might suggest that the subspace relays allow for faster
short-range subspace communications signals which can travel much faster and is lag-free vs the slower
long-range subspace communications signal?
I do hope that there are faster subspace signals though. A 52,000c subspace signal would mean that a subspace sensor signal would travel only 100 AU in a second which limits how far forward nav sensors can look. A ship traveling 21,000c ("The 37's") would be moving 43 AU/second which would make traveling at high warp a very near-sighted flying experience.
"Lift Us Where Suffering Cannot Reach"
FIRST SERVANT: Do you know the speed of propagation of subspace radio signals in long-range communications?
SPOCK: I believe it is roughly 52,000 times the speed of light.
FIRST SERVANT: Really? Wow, that's... super slow. No wonder you have to use subspace relays. At those distances, the signals would degrade long before they were received, even when radially polarized.
SPOCK: I do not know many your age who grasp radial polarization.
FIRST SERVANT: I'm interested in it because I thought it would be fun to have a friend across the galaxy. I once worked on my own subspace frequency. I bet I could generate one by rewiring this biobed.
"The Serene Squall"
PIKE: How long for a request to reach Starfleet?
SPOCK: At this distance, two days. This sector is too far from Federation subspace relays to provide rapid communication.
...
PIKE: We'll drop subspace buoys as we go to maintain comms.
SPOCK: Those buoys will be our only connection to civilization.