Webinar won't appear in my integrated calendar + I can't add it to the prepared shared calendar

Correct for meetings you can integrate your email calendar and your scheduled Zoom meetings would appear there. Webinars however are not integrated directly to your calendar. You could create a calendar event for each webinar and add it to your shared calendar. If you don't have too many webinars this won't a be a big deal. It sounds like you don't want to be making a copy just for yourself though in which case you may want to explore using APIs to add an event to your integrated calendar whenever an webinar is created. That way you would have a centralized calendar display where you and your team can view upcoming webinars. There is also a Zoom integrated app, Salepager, that provides a calendar display for webinars.