Apps I don't care about won't delete from my Watch. Am I stuck with them?

Like most smart devices there will always be some apps that are native and tied to the OS. They aren’t forced on you (meaning you aren’t required to use them in order to have your device work). Yes they exist but you aren’t forced to interact with them.

As far as deleting….it appears that the cycle tracking app can be deleted. I tested it by going to the app grid, putting my finger on the cycle tracking app until all of the apps started to jiggle, and seeing that the app included the x which means it was delete-able. I don’t see an actual app for handwashing but if you for some reason don’t like the countdown to start when you wash your hands you can toggle that off in settings.

If they are standard Apple apps such as reminders or weather they cannot be removed. However if they are apps you’ve downloaded such as games etc then click on your Apple Watch app scroll down to the bottom and it shows download apps to you watch. Once you click then it gives you the option to remove them. Hope this helps ☺️

Some native “standard” apps can be deleted (such as Cycle Tracking that the OP was referring to). Cycle Tracking is an app that came on the watch automatically and did not need to be downloaded.