There is code for prioritisation yes, but there’s more-or-less no overlap with what is needed for Network Auth. For example, things like form field id and name matching play a part for web forms.
You should be able to get the web form priority right by ensuring that the web form entry has accurate form field information (such as id and name) for the username and password fields. The easiest way to do that is let Kee save the entry for you but if you don’t want to do that, you can edit the form field information directly within the entry, albeit only after inspecting the page source code of the form (fine if you’re technical - if not, see what I said about getting Kee to save it again as being easier )
Manually setting the priority field is another possible semi-workaround, as you say but honestly I’m not that comfortable with the feature (it’s way too hard to use it correctly, especially over many years as memories fade) so it might not survive future Kee updates, especially if I can make improvements in other areas to reduce the need for it even more.
I’ve not tested this (and can’t promise it will work in the long-run so don’t go overboard spending lots of time on the configuration of many entries) but you might be able affect the order of the Network Auth results by using the KeePass entry ordering feature. So if you have the Network Auth entry higher up in the list of entries in KeePass but more accurate form field matching on the web form entry lower down the list you could get both working as you want.