Alarm on iPhone and Watch aren’t fully synced

Yes I understand and good luck with that.

The only way to have a silent alarm (or anything else) on the Watch that I have found is to turn off the sounds in Silent mode. (Red bell in Control center)

My advice remains

If you want an alarm to sound on the Watch, set it up on the Watch. If you want it silent turn on Silent Mode.

If you want an alarm to sound on the iPhone, set it up on the iPhone where you have the choice of sound including a purchased silent tone.

If you want to have the alarm on the Watch when it is set on the iPhone (the iPhone has to be closed for it to reach the Watch) then turn the Push Notifications back on but turn on Silent Mode on the Watch

Other advice may come along, but if you think this is a bug you can report it here https://apple.com/feedback

Change the setting, it is not a bug

iPhone, Watch app, My Watch tab, Clock, Push alerts from phone, Off

If you want an alarm to sound on the Watch, set it up on the Watch. and vice versa.