I also understand that it is easier to clean.
Not sure why that would be. Where did you get this idea from?
why isn't the CX style bikes more popular?
But they are! Or
were...
To pick one really popular example, the Specialized Tricross was marketed as a CX bike but was really more of an all-round workhorse, for exactly the kind of use you're interested in, but they don't make them any more. I suspect very few people used them for actual CX racing (true CX racing bikes won't have mounts for panniers or mudguards).
There are better options for commuters these days than those pseudo-CX bikes - the Tricross has effectively been replaced in the Specialized line-up by the superb Diverge, which is more what you'd call an 'adventure' or 'gravel' bike.
As for tyres, I think 28mm with small water displacement grooves on will give me most confidence. The cyclocross I sat on today had 28mm tyres with little nobbles and it looked fine, however when I have a quick look online I only see cyclocross bike with no smaller than 32mm tyres - maybe this cyclocross today had special tyres on.
Knobbly CX tyres are rubbish for riding on roads and I can personally vouch for this - my last off was purely due to lack of grip when riding with knobbly CX tyres on slippery tarmac. Slick tyres are much better. And by the way, those grooves are purely cosmetic and don't serve any useful purpose (the round contact patch of a bike tyre means water is adequately displaced).
As for size, that's up to you, but many people prefer 32mm for the superior comfort. And they won't necessarily be any slower than a narrower tyre. My all-time favourite commuting tyre is the Schwalbe Marathon Supreme in 32mm - fast rolling, very grippy and reasonably puncture-resistant.