If you play video games 3 hours per day every day from the moment you are born to the moment you die and every joule of it comes from fossil fuels, you still produce fewer emissions than a single private jet flying one time from New York to London.
0.5 kW x 3 hours/day x 365.24 days/year x 80 years = 44 MWh in a lifetime of gaming 3 hours per day.
300 gallons/hour x 7 hours x 47.4 kWh/gallon = 100 MWh from a single flight.
Also, a person running on a powered treadmill produces more emissions than a person playing video games on a computer:
0.7 kW treadmill + 0.5 kW person engaging in exercise = 1.2 kW power consumption on a treadmill.
0.5 kW gaming PC + 0.2 kW person sitting = 0.7 kW power consumption.