Andy Molar and Summerfield build beautiful road racing machines. If they can survive, there will always be somebody making tyres in sizes to suit their bikes - 18 inch and 19 inch. There was somebody advertising tyre for vintage machines a while back. What sort of idiot would make Fullauto cylinder heads for Commandos ? - the market must be miniscule. But they are available.
About tyres going off. They go hard but after a couple of laps of a race circuit, they usually soften. But one thing I never do is let a bike stand with a flat tyre. The stress on the side-walls can cause them to crack. And you might not detect it.
Many riders are tyre dependent, but mostly tyres are about rider confidence - if your tyres are new, you feel better.
If you go out onto a race circuit and don't use the first 5 laps to come up to speed, you risk dropping the bike. Even when tyres are 20 years old, when they heat up they will regain some grip - a lot depends on tyre pressures. If you don't have enough grip, you can always lower the tyre pressure. But if you have really old tyres, you are better to replace them and start anew.
The other thing is the steering geometry on your bike. If you have enough trail on the steering, the bike stays more upright and oversteers in the correct direction as you accelerate. Then you do not need a lot of grip. But more grip is always better.
Back when I was a kid, there was a lot of bullshit involved in racing - we often wondered why the old guys were faster, on old bikes which looked like shit-heaps.
What we really needed to do was test-ride their bikes.
I have ridden in practice sessions with old A grade riders. They were not much faster, but much smoother. You need a bike which helps you, more than you need increased horsepower.
When you have a new race bike, it can take several race meetings before it become sensible. It is easy to build a bike which will scare you shitless.