Which kind of strategies did you guys attempt? And how long were the races? If you have a good strategy and pit in time, you should never have a puncture.
Far too many of your league were running the soft tyres far too long but this is probably down to people not doing setups/being inactive. With the recent update soft tyres just aren't going to last 13 laps.
Take a look at the stratgeies of the leading two in your rookie, they only did up to 7 laps on the soft tyres and did extra stints, also look at the higher pro and elite leagues, some used extra stints on softs or did 1 stops on the hards which wear slower.
As for puctures, they are just a chance thing. If you use your tyres too long that they get too low on health then you are likely to get a puncture, but it is just up to chance/luck, some people will get a puncture while others wont.
I have seen a few punctures in our league as well but this was by the same manager who appeared to be running heavy fuel loads when was pushing hard for both drivers. This was also due to him not being online either.