There's definately a big jump up from proper cheapo tyres to competent budget ones. My current tyre came with horrible dealership specials on and sure, it could be driven safely but only by being very conservative. And stopping distances, margins for error etc all badly hit, especially in the wet.
So I've just stuck some Hankook Ventuses on, at £110 a corner. Hardly top end but the difference between that and the Autogrips it had before is ridiculously huge. So that translates into more safety for sure, but also less need to baby the car all the time, which is also good.
That said, I used to have a lot of fun with my old Focus and its feeble tyres, it'd find its grip limit very easily even at fairly low speeds.