The words “all faiths and none” have entered usage as a set phrase - a bit like those Shakespearean or Churchillian ones that are always said in that order.
It’s a bit like saying ‘last but not the least’, ‘without let or hindrance’ (let means hindrance), ‘law and order’ (nobody says order and law) and so on and so forth!
I confirmed this by doing a quick web search on the phrase:
You’ll find a great many web pages with this phrase in their title or even part of the URL. I even found an acronym definition for it - AFAN!!
Blimey, even the Church of England have an ‘afan handbook’!!
https://www.churchofengland.org/media/1123577/afan teachers handbook.pdf
So the reason ‘people of all faiths and none are welcome’ works is because we (English-speakers) subconsciously take in those words as a single semantic unit - As if you could say ‘we are AFAN+’!
If you’ve done any computer programming, you might notate it thus:
(people (of all faiths and none)) are welcome
Only when you starting dissecting the actual wording of the original sentence used above ignoring or oblivious to its idiomatic history do you start doubting its grammatical validity.
I would still take on board the objection @KIKC had about ‘… none are welcome’ - maybe you could say ‘We welcome people of all faiths and none’ - after all this is better expressed actively rather than passively.