Yes, a shorter sidewall tends to improve response time in a tire - the time lag between your turning the steering wheel and the vehicle responding.
HOWEVER, it is possible to change the response time by stiffening up the sidewall, so even tires with taller sidewalls can be made fairly responsive.
And F1 tires? Those are MANDATED and everyone has to use the same sizes - which has not changed for quite a while (I'm guessing 20 years). What has happened is that F1 regulations froze things so there wouldn't be such a difficulty in swapping between tire manufacturers - and then took it a step further and specified a single supplier.
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