The four professional sports leagues all have Bay Area teams. The San Francisco 49's technically have their stadium in San Jose, because land is so expensive in San Francisco. Obviously, the San Jose Sharks are in San Jose. The Warriors and Giants are in San Francisco proper, but for how much longer? When they build new stadiums, the question will arise why are they using such expensive land for stadiums.
At that point, Oakland will be a strong competitor for the new location of stadiums. They have the cheapest land in the area, they have a central location, and they have a government willing to bend over backwards for the stadiums. In the past, they also had teams there already, which made placing San Francisco teams there difficult, but not anymore.
So the answer to your question is that there is every reason to move a professional sports team to Oakland, but probably not much reason to move the brand name to Oakland. The "San Francisco" 49's might move from San Jose to Oakland, but still use the San Francisco brand name.