Well I can attest that a thicker radiator does cool better. I recently replaced my perfectly good fairly new stock replacement radiator with a large Be Cool radiator and behold... I thought the car would never reach 180 degrees. I was getting worried it would not warm up all the way, but finally it did in fact warm up to 180 after 45 minutes of driving in 70 ish degree weather. With the stock radiator the temps would stay in the 210+ range all the time, and then when you put the A/C on it would try to overheat every single time and still does. My A/C condenser is probably in need of replacement. I know my stock fan is working on both high and low. I have two stock fans, both work.Very little, but you can squeeze it in there by removing the dome cover from the fan motor. That gives you like a quarter inch.
The best advice I can give you is to just use an OEM replacement radiator.
The “thicker radiator” is just a marketing ploy, it won’t cool any better. You need more airflow, faster coolant speed, and lower fan set points in the tune.