Hello XHelation.
Thank you for using Apple Support Communities. You mentioned that you're unable to hear your alarm when wearing your wireless headphones. If so, we'd like to help out and see if we can recommend some steps you can try. According to the following resource here: How to set and change alarms on your iPhone - Apple Support
It states the following:
If you connect headphones to your iPhone, the alarm plays at a set volume through the built-in speakers on your iPhone as well as wired and wireless headphones.
Restarting your device and repairing your headphones was a great step to attempt. There's additional steps you can try to further isolate this issue. First, make sure that you install any pending updates available for your iPhone. Keeping your device fully updated can help resolve various issues. Just in case, this resource can guide you through the update process: Update your iPhone or iPad - Apple Support
1.) Back up your device using iCloud or your computer.
2.) Plug your device into power and connect to the internet with Wi-Fi.
3.) Go to Settings > General, then tap Software Update.
4.) Choose the software update that you want to install. You can choose to stay on iOS or iPadOS 15 and still get important security updates. Or you can choose the option to upgrade to iOS or iPadOS 16.
5.) Tap Install Now. If you see Download and Install instead, tap it to download the update, enter your passcode, then tap Install Now. If you don't know your passcode, learn what to do.
You can also test this issue using different headphones as well. If issues only occur while using one specific headphone, check to see if there's available firmware updates for your headphones. We hope this information is helpful. Cheers.