With the Storm Covenant gone and the Banished coming to the fore it really is looking like we’re seeing the final shift away from the Covenant as these religious zealots. I mean, if the only two note worthy force in the games are the Swords and the Banished then it means that the religious identity of the various races is fading? But I am not so sure.
Reading the Phoenix logs of the elite honor guard, he does note that he has abandoned his parents who still cling to the old ways. This would, possibly, suggest that he is identifying himself as a bit of an outcast and that the majority of elites still adhere to the old ways.
We also have the Tales from Slipspace comic which shows the elites killing large numbers of regular religious brutes before coming into conflict with one of Atriox ships. Implying they are in the minority otherwise they would have encountered more Banished before then.
You then also have Decimus, ostensibly Atriox right hand man, but who himself still believes in the old ways and venerates the Forerunners. If your second in command doesn’t fully buy into your factions beliefs then it suggests that many people are following Atriox out of respect for him as opposed to buying into his secular beliefs. In Halo 5 on Sanghelios you also hear soldiers in awe of the Guardians and clearly having some doubts or ill will towards what the Arbiter is doing; they just rightly recognize how awesome he is. So although they say that the Brutes look to Atriox as a strong leader this is very much about personality and not ideology.
I think you also have to consider the fact that the Storm Covenant was ever a thing. Both the Swords and the Banished are really not characterized as popular movements, they’re simply extensions of one awesome charismatic warlord. Atriox for the Banshed and Arbiter for the Swords. The Storm Covenant had Jul M Dama, who as we’re told “was no Prophet” and who was pretty consistently portrayed in the games as a bit oafish and incompetent. The fact that so many people joined his cause and continued to fight after the mans death. We’re point blank told that if the Arbiter died then his faction would die with him. This was not the case for the Storm Covenant, suggesting that it was a popular movement based on ideology rather than one centered on a charismatic leader like the Banished or Swords.
So I think this suggests that the majority of the Covenant races still zealously buy into the practices of the old Covenant.