After you did that, did you mow with it for over the 30 min time limit to see if anything changed? If not, try mowing or running the motor for the 30 min to see if it still shuts off. (If) and when it shuts off, loosen the gas cap and listen for a vacuum release. If you hear one a new cap is in order, if the cap is plugged (and it still could be after you blowing into it) the fuel pump draws fuel from the tank to the carb until there is a vacuum build up in the tank above the fuel level which the fuel pump can not overcome to draw fuel to the carb. That is when the mower actually does run out of gas even though the gas tank may have 3 gallons of gas in it. A new cap runs about $10. Next time the mower shuts off remove the cap and try to re-start the mower. It will take some cranking as the fuel line and filter will need to re-fill with gas if the cap is the problem. If it doesn't run after this procedure then, like you said, the cap is not the problem and more testing is in order.
A bad cap may be the reason the mower needed a new fuel pump. The vacuum eventually breaks the fuel pump diaphram which causes a gas leak at the pump.