Right, he PRESENTS a budget, but Congress is who writes it because Congress writes laws and the budget is a law.
Who do you think wrote the 2022 Omnibus Budget Act?
The budget is largely written by the OMB from presidential guidelines. Congress must pass this budget but the final product is usually within 5% of that presented by the president. It is much too long and complicated for Congress to actually do most of the work; and, 70% of the budget is mandatory spending entitlements (Social Security, Medicare) which Congress does not determine annually.