"it requires me to pick a specific view from the ARCH Link"
Yes. Because dimensions, text, linework, and all other annotation objects are view-specific. If you worked exclusively with an architect who always, always, always, had identical view naming conventions you could adjust your template in the way you describe, with a separate template for each discipline on each floor by creating a placeholder ARCH model, linking it into your template, and choosing the linked views you need. I would strongly advise against doing this though, because that prerequisite will never, ever happen unless you work for a very unique firm.
You can adjust the linked view/model visibility setting apart from your template - I'm guessing your views have a template linked to them, and all of the V/G settings for model objects/annotations/linked models/etc. are gray when you open the VG dialog. To fix this, in your view properties, find the "Template" line (the value will likely be "HVAC floor plan" or whatever you named it), and change it from the template linked to the view to "none". This won't reset the VG overrides, it'll just make them editable.
In the future, when setting up your model, instead of choosing "Apply Template to View" choose "Apply Template Properties to View". This gives you the basic override settings from that template while keeping them editable so you can customize the view to the project as needed.
