That is not true. If it approaches with less than escape velocity it will gain all the velocity necessary to escape in the process of approaching the Sun. You

That is not true. If it approaches with less than escape velocity it will gain all the velocity necessary to escape in the process of approaching the Sun. You could think of it as speeding up as it falls toward the sun, it then slows down by the exact same amount as it leaves the Sun. In order to stay near the sun it needs to lose some of that speed, and given that momentum is conserved, the only possible way is to either hit the Sun or send that momentum to a third object.