I have ABS and Audi Brake Assist and I will STILL rather spend the money on a good quality tyre and and not some budget cr@p.
If you transport your family in the vehicle it becomes more important.
Two most important things that need good quality components are tyres and brakes IMO. They mean the difference between life and death, and the money saved buying budget tyres means squat if you're freaking dead.
I think there needs to be some distinction made here.
You get budget aka super cheap noname brand like tyres, then you get middle ground "compromise" tyres often made by the same manufacturer but for the buyer who wants to pay less for largely the same thing (like VW & Audi exist side by side with 95% the same cars) and then you get premium aka expensive tyres.
The budget tyres are not great and probably a 20% loss in performance over the premium brands.
The compromised ones on the middle like Hankooks and Achilles are probably 5-10% off the Premium brands performance wise but probably 40% less in price.
Ultimately it comes down to doing your research and not just walking into the shop asking for the cheapest ones they've got because they are the cheapest. You want to strike a balance between cost, performance and longevity.
Depending on your driving habits you might very well end up with a premium brand manufacturer but NOT the performance version of the rubber because you don't need it because you want them to last longer.
In the same way the performance rubber from a lesser brand could very easily trump the economy rubber of a premium brand and still for less money.
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Then of course there is the seat of the pants test. My car came with Hi Flys that I've never heard of but I drove it hard and quite happily and never had a problem with them. That being said if I saw them on the shelf not having driven them I would never have touched them due to lack of any information that I could find on them.