It's an easy game to play actually. Strict contructionists will only recognize discourse that can be understood in 1790, or whichever relevant time. They use dictionaries from that time and the writings of the amerikan founders to make their points. You won't easily find anything from that era that implies "religion" is anything other than Christianity and it's various sects. To assert otherwise would be to legislate without congress. So they can argue that excluding non-Christians and non-Protestants is in line with the intentions of the authors regardless of article 6.

Is it a perfect line of thinking without contradictions? Of course not, but neither is the counter idea that America was designed to accommodate non-Christians.