The company probably owes U.S. and state corporate income taxes because income from services performed in the United States are usually considered "effectively connected" with the United States.
The fact that the servers are located in the U.S. is pretty much irrelevant, relative to the fact that the services are performed while located in the United States. I can't think of a single tax case that has ever turned on the location of the servers in a company.
Unlike a U.S. company, a foreign company is not taxed by the U.S. on its worldwide income, nor is the individual, a non-resident alien (having an F-1 visa rather than a green card) taxes on the individual's worldwide income. But, a non-U.S. person is still taxed on income that is effectively connected with the United States.
Generally speaking income from property is not effectively connected with the United States merely because it is managed by someone located in the U.S., so if the company had owned an apartment in Brazil that it received rental income from, for example, that would not be subject to U.S. taxation.
Also income from intangible property (like interest payments on loans or dividends on publicly held stock) is generally not subject to U.S. taxation if paid to a non-resident, non-citizen of the U.S.
But, generally speaking, income from the performance of services is taxable in the place where the services are performed. For example, Colorado can impose state income taxes on income earned by a Texas baseball player while playing at a stadium in Denver.
The lack of a salary or employee status shouldn't change the fact that the income received by the company from performance of services in the U.S. is effectively connected with the United States. When the owner performs services in the U.S., the company is performing services in the U.S. and so it is subject to taxation in the U.S.
Dividend payments from the offshore company probably wouldn't be subject to U.S. taxation in this scenario, but the company itself would be subject to corporate income taxes in the U.S. from the profits it earned from the services performed in the U.S.