Yes!
> That's why you're allowed to deduct it
It's complicated.
First, Sales taxes cannot be deducted from SALT if you deduct state income tax. You have to pick which one you deduct. But if the intention was to avoid double taxation, they would have allowed you to deduct both.
Second, SALT exists from the era when top marginal rates were 90 percent, so you could end up in situation where you paid more than your income in taxes. Actually this is true even with the SALT deduction, but the deduction was basically a tax break to top earners in order to compensate for very high federal rates. The rationale for it doesn't exist today.
Third, double taxation does not refer to the amount of tax paid but to the fact that the same income is taxed twice. Deducting the sales tax (or the income tax) would not eliminate double taxation, it would only reduce it by a small amount.
E.g. if you earn $1 in income and spend half of it, and if the sales tax is 10% and state income tax is 10% and the Fed tax is 30%, then:
* Without SALT, your liability is 45 cents (thank God you only spent half!) and you have triple taxation on $.50 (sales, state, federal) and double taxation on $.50 (state, federal)
* With SALT, your liability is 42 cents and there is triple taxation on $.50 (sales, state, federal) and double taxation on $.40 (state, federal). Thus the SALT deduction saves you 3 cents by reducing your federal liability from 30 cents to 27 cents. But you paid 15 cents to the state. This is because the SALT deduction is not a tax credit, it's a tax deduction that just reduces your taxable income a bit.