This is a known pattern rather than something you have misconfigured, and macOS behaves differently enough from Windows here to be worth explaining properly.
Safari supports the Gamepad API but historically it has been the strictest browser about the user gesture requirement, and it is pickier about window focus than Chrome is. On macOS specifically, a Safari window that is not the frontmost window, or a page you have not clicked into since it loaded, will often report no gamepads even after you press buttons on the controller. So the first thing to do is click directly on the page body, make sure Safari is genuinely the frontmost application, and then press a face button. That alone resolves a large share of these.
The second common cause on a Mac is something else holding the controller. Steam on macOS claims controllers quite aggressively and does not need to be in the foreground to do it, so if Steam is running, quit it fully, including from the menu bar, then retry. Any other input utility counts too.
The third is the macOS build. Support for the DualSense specifically improved substantially across macOS versions, and on older builds the controller pairs and shows as connected in Bluetooth while never being exposed as a gamepad to a browser. That gives you exactly your symptom: visible in Bluetooth, invisible to every browser. You are on 15.3, which should be fine, so this is less likely for you, but it is worth confirming you are fully updated.
For testing purposes I would use Chrome on the Mac rather than Safari. Not because Safari is broken, but because Chrome's implementation is more forgiving about focus and gesture, so if the controller appears in Chrome and not Safari you have learned it is a browser behaviour rather than a hardware or pairing problem, which is a useful thing to know in thirty seconds.
