I don't know about the servers but to my knowledge, Airplane Mode does not reset the network. To do so, you have to go to Settings>General>Reset and tap on "Reset Network Settings".
Siri performance has been hit and miss for me today too. It has such potential. Apple needs to get the "it just works" bit installed, then I'm sure it will be more reliable.
Just started happening here for me in terms of connection problems. It was fine at 2 now it's 4:20. Haha.
Anyways, childish jokes aside, it is most likely due to Apple Servers and maybe doing upgrades to the software as it is still a BETA feature. Who knows why Apple decided to make an ad for it as it isn't amazing and BETA.