what is the logic behind the IRS allowing a choice of one state tax to
be deducted on Schedule A (state income or state sales tax), but not
I believe the idea was to allow deduction of income taxes only, however - there are States that do not have income taxes. So in order to not discriminate against taxpayers living in these States, sales tax deduction was allowed as a choice, if it is more beneficial.
And why does it specifically exclude certain other state taxes as
well? (such as a tax on gasoline, car inspection fees, assessments for
sidewalks or other improvements to your property, tax you paid for
someone else, and license fees)
Property taxes are deductible, as long as they're based on the property value. Flat assessments are not based on the property value, thus are not taxes - they're assessments. Assessments are added to the basis in the property and reduce your gain (and thus reduce the capital gains tax, not income tax).
Portion of the car inspection/license fee that is based on the car value is deductible, as a personal property tax. This portion (if exists, not all States have that) should be marked on the DMV statement you get.
Consumption taxes (with the exception for sales tax, as described above) are not deductible.