@boag The telecommunications thingy is one sided unless you set up multiple ones, hence it's not as effective for communicating with specific individuals, plus they're rare (the transmitter, that is, not the receivers). Context would show you this since the past king was using the devices for major announcement rather than distributing it for logistics (Poncho could have carried one while flying but did not, they were told by letter about the landslide in the god forest, etc..) and that the only ones that appear to have one are the king + 3 major powers.
For obvious reasons, the dukes would've been aware of the circumstances on how Souma came to the throne and had various means to check on the authencity (hence why Excel had him spied on). The other dukes simply chose different conclusions. The sky duke also is not against Souma but rather is simply loyal to the lion duke because politics.
Using it for long distance comms isn't the best idea at the time since the mc had a: ruined kingdom to fix, assumed that the dukes were loyal and it required everyone needed to be present at the time.
Using a minor person to relay info isn't the best idea as well since the issue of possible incorrect information being delivered (which would be disastrous for a non-portable real time transfer of information)
Also while the idol shows aren't the best idea per se, uplifting the masses in times of turmoil (and distracting them) is never a bad idea. Particularly if you're mass evicting the nobles, which would have alarmed the common folk if suddenly their lord goes missing.
PS. The line "common sense like us" sounds stupid to me. The strength of this series is that the characters have their own set of philosophies and there isn't a perfect answer, which shows from how the major political players move. So saying that "I with my common sense would do x" is the stupidest idea since even in the real world, implementation of simple things don't necessarily agree with how the majority think.