Whilst that's all good standard practise it shouldn't prevent traffic being delivered (he shouldn't need to press it 10-20 times).
What might be an interesting test - if you haven't changed things already - would be:
Turn on just one of the WiiM devices, the furthest from the Hue hub, and see if everything works.
Assuming it does, turn on another, ideally in the same place (they don't need to be connected to anything) and test again, and continue doing this until you have them all on.
This would test whether it's broadcast related.
If everything still works then bring one closer to the Hue hub and test again, although this should only slow things down.
I know they announced in the other thread about the ability to disable features on the mini to conserve memory, but it might be beneficial for network traffic too on the rest of the product line.
I know UPnP, Squeezelite and Chromecast all announce differently, but I don't know if the rest do or whether it's all just additional services under a single mDNS broadcast.
EDIT: Sorry I should have realised that ZigBee didn't use WiFi so it could be interference that's blocking the ZigBee mesh.
The tests as outlined above should still help identify the source of the issues though.