State inspection and vehicle registration is two different events.
The state inspections in Texas for a car that old is pretty simple (25 years old?). They check the lights and horn, drive around the block and stop quickly, and by that, evaluate if it is 'safe to drive'. At times they have actually looked under my TR6 hood (you are supposed to be running an air filter and any pollution devices originally installed). There is no emissions tests. You must have proof that it is insured.
Registration as a classic may negate the need for regular state inspections (I'm not sure exactly). But special registrations also limit drive-ability. If you are registering a car that came from another state, you must first have it inspected at a state inspection facility -they want to make sure the title numbers match the numbers on the car.
My 74 TR6 is not registered as a classic, because I want to drive it when ever and where ever I want, and I don't mind getting it inspected every year. Classic and antique registrations have different rules, and insurance differences. It gets involved.