Not really, Ferrari's budget is even higher than Mercedes, Red Bull is around half and still wins.
One of the new rules is to control Wind tunnel runs. Factories usually run wind tunnels 24/7, mercedes and ferrari run 2 wind tunnels. 2021 the maximum runs per week are 44, and will change according to team success. Mercedes will only make 36runs per week (90%) and Racing Point or other teams worst qualified will run 47 per week (112.5%)
Slower cars will catch up in 2022-23. But don't expect them to win, main teams still have a newer version of the engine. Overall, it will become more competitive next season.