What you describe is called a "race to the bottom", and it's generally Bad for the citizenry.
I imagine you'd agree that an important exception would be, for example, when it's a race to the bottom of prices on consumer products. Yet that bottom is a floor beyond which it is difficult to lower prices and still make a profit. That floor is rarely zero.
The states are in a similar, but not identical, position. They want to attract individuals and businesses by lowering the tax burden on them, but must provide a service of, say, enforcing the law to protect the individuals and businesses from thieves, which costs money. If the state lowers the tax too much without becoming more efficient, it will fail to adequately provide its services (e.g. crime will go up), and it will be unable to make a "profit", i.e. attract businesses and/or individuals, who do not want to live in a crime-filled world.
Unless, of course, those individuals and businesses simply find adequate non-governmental ways to replace the functions of their government, in which case, the lack of tax revenue is irrelevant.
The end game is nobody pays taxes
Possibly, and if they are not necessary, why isn't that a good thing?
But if taxes are necessary, then perhaps the state which uses its tax revenues more efficiently and is thus able to provide services of a higher quality wins out in the eyes of prospective businesses despite a higher tax rate.
10.0 times 0.1 is hardly ever 1.0.