We also don't bring into our home any allergen that could be easily aerosolised and/or stick to cookwares - no nuts, peanuts, or wheat/potato flour. When my husband cooks anything with wheat or potato (pasta and perogies only, no flours, and it happenns only a few times per year, not daily) he makes sure to clean up everything, scrub the pots, and then put them in the dishwasher for an extra clean. That way, our kitchen remains a safe place. We do not use separate cookware as a result. We also only use stainless steel pans, nothing with a coating. (We have dedicated allergen-free cast iron cookware, mind you.)
If you use separate cookware you may also have to store them separately and clean them separately too, not to mention traces that could be on utensils, counters, cutting boards, etc. I think having better allergen control measures, such as limiting allergens in the house, thorough cleaning routines, and food storage segregation would be more important than separate cookware, as separate cookware alone, without those other measures in place, would offer a level of false security since traces could be introduced elsewhere.