This has been most helpful.
Since getting - totally out of the blue - a new Motorola modem/router, turning it off has caused startup problems. They match the above scenario completely.
Our new deal, AT&T's U-verse, seems okay, but the fitting of a 'Whole House' filter caused some hassles. That's by-the-by.
As an old electronics bod, I've seen too many bits of kit go up in smoke to want to leave cheaply made odds lying around - especially in a wood-framed house. Here, in the Texas drought, my fireman neighbor is never home.
Some of my friends go on holiday with little green lights twinkling away all over the house. It freaks me out.
For now, it's 24/7 and no problems, but if I'm away overnight, then it's that cup of coffee until the new unit has become sapient.