The only people who really know the answer are people who are at least as heavy as you and who ride on 23mm tyres. Personally, being 6ft 6 and usually between 20 and 21 stones, I know I have never trusted typical road bike wheel and tyre combinations, which I think of as flimsy. What I did from the outset was to go for 28s and the additional back-up of Slime (self-healing) inner tubes. I also had the spokes in my rear wheel (the main weight bearing one) upgraded to "Alpine 3" triple-butted spokes. There are many on here who would consider that to be overkill, and of course you are a good 3 or 4 stone lighter than me, so it is a question of degree just how far down that line you choose to go.
What I can say, though, is that in the last seven years I've only had one puncture and one broken spoke out on the road. The puncture was entirely my own fault for riding over a huge sharp rock in the middle of the road instead of looking where I was going. So really virtually no problems at all, despite taking on gravel paths and bumpy, poor grade roads with regularity, and covering nearly 14,000 miles in all. During the last 2 years alone (in 50+ club rides) I must have had to wait a dozen times for clubmates with flat tyres or broken spokes on their standard road set-ups. I now know I would make all the same decisions over again.
