Also not true. Footwear companies do not give county cricketers full sponsorship; they offered them discounted rates, but it does not tie them to the brand.
You will will iften see players using 2 or 3 brands in the same season.
Overall you are right, unless of course their bat sponsor happens to make shoes. New Balance, Adidas, Puma etc.
The main reason Plunkett left GN to go to NB was the offer of as many customised bowling shoes as he needed.
County pros get a discount on whatever brand the PCA is associated with at the time, but if they don't like that brand they will have to shop elsewhere or rely on their bat sponsor to buy them for them. We provide shoes to our Focus sponsored professionals, and in fact provided shoes to Rob Sayer at Leics before Focus was a thing in the UK. We've also looked after a few H4L players at Kent when Matt has asked us to sort a deal for them.
You see a lot of players now with a bat sponsor also tying up a deal with a retailer. The most notable recent example is probably Duckett and Pro Direct. Great deal for them as they managed to get him to have Adidas' cheapest shoe by the look of it.