Okay, turning off the "Auto Network Switch" worked like a charm on my Note 3 (on the original Android 4.3.) Wonder why ...
I read that option to mean that I want the phone to use the carrier for data if WIFI is not available, not if and when there is no internet connectivity or it is "slow" - whatever the criteria for "slow" are here. It seems we are not told, let alone given the option to adjust this.
I'll have to watch now if (and when) the phone switches. Indeed it almost looks like there is a carrier induced scam going on that says "You aren't buying enough data from us, so we'll teach you ..." In fact, I frequently get scary SMS's towards the end of the month offering me a bigger data bundle.
On the other hand, this only started with my new router - the old one was fried by lightning. But on the new router all my other equipment is working fine - regardless of WIFI or Ethernet cable connection.
One other strange thing: From the moment I connected after turning off the Auto Network Switch all web links wanted me to enter my userid and password - even the advertising and analytics sites that are called by practically all commercial web pages all the time. Fortunately that nonsense stopped after I power cycled the phone. I wonder what is going on here.