At 2000 miles....you are starting to get flats frequently because you have hardly any tire left. It's not really the tire's fault. It's worn out.
I'd be EXTREMELY nervous about anyone riding a tire with the 4000 miles you had on your old tires. Not for flats...for control, cornering, braking and the possibility of blow outs at speed.
Please don't try to set mileage records on tires
That said...The Continental is a nice replacement tire.