I've seen there's Canadian gadet for than - firm plastic base with metal crown-like spikes that you fixating under your boots. But they're expensive - about 90 bucks. I tried cheap from China, but they're not firm, they're soft made of silicone and my shoes they like dress shoes (like oxford or brogue type), they don't have thick rubber base as big boots or sneakers, so I felt those cleats with my feet, walking was uncomfortable.
Maybe you need a two-step approach. I don't have a brand to recommend. I have a technique you might find helpful.
Keep your cheap cleats; but, put them on over a pair of rubber overshoes which will stiffen the soles and protect your polished shoes from wear and tear of the cleats rubbing against your dress shoes.
Step 1. Shoe protection
Step 2. Put your anti-skid ice cleats on over these galoshes. More complete weather protection for your footwear. Stiffer soles for your foot comfort.