According to Engadget, phone coolers are not snake oil, but they solve a problem most people don't have. The short answer is that most users don't need one. Active cooling can prevent heat from degrading sustained performance, but typical phone use doesn't generate enough heat for long enough to trigger throttling.
The article distinguishes between basic air-circulation models, thermoelectric Peltier coolers, passive cases, and cooling apps. Only active cooling can physically remove heat. Engadget cites Notebookcheck's 3DMark stability scores for the ASUS ROG Phone 8 Pro: without the AeroActive Cooler X, scores were 77% and 79.2%; with it, they rose to 92.7% and 90.8%. So a cooler can reduce lag if heat is causing it, but it won't fix poor Wi-Fi, server lag, or a game the phone simply can't run well.
The cases where a cooler makes sense include long gaming sessions, livestreaming, video processing, outdoor summer photography, and phone use in cars while charging and navigating. Before buying, Engadget suggests trying an AC vent, removing a thick case, avoiding direct sunlight, and not charging during demanding tasks. Wireless charging runs hotter than cable charging. If throttling persists after those steps, a dedicated cooler is justified; otherwise it adds weight, fan noise, cables, and grip interference.