Personally, I fight against the idea that it's an alternate timeline because one of the biggest reasons I like the show is because it is the same characters that I fell in love with decades ago.
Strange New Worlds actually helps me appreciate the characters even more than I already did by filling in their backstories and giving them far more character than TOS ever could have gave them in the 60's.
Uhura didn't exactly get much character development back in the day. But now, thanks to SNW, I can appreciate and look back at her personal journey to get where we saw her in TOS. Same with Spock, Chapel, etc.
Strange New Worlds has done nothing but improve The Original Series characters and setting. To remove it from the prime timeline would be doing those characters a great disservice.
I completely understand your viewpoint. I see things a little differently. I feel like, despite my absolute love of TOS, that it is quite dated and that because it's a product of its time, that there's no need to 'update' it or 'improve' it with some 50-years new future production that claims to say it takes place in the same continuity even though there are quite jarring fundamental differences between the two productions. It's like saying that the Christopher Nolan Batman films take place in the same continuity as the Adam West '60's TV show. No one in their right mind would think that was the case, even if suddenly Nolan were to say that it was so.
With that said, I like SNW and agree that they are making these characters much better than they originally were. But I simply don't see them as the same characters as their TOS counterparts, just like I don't see TOS BSG Adama as the same guy as nuBSG Adama, because they blatantly aren't, because nuBSG is a reboot. But nuBSG also took the old characters and made them better in the reboot format, and that's pretty much how I feel about SNW. It's a reboot, despite what CBS/Paramount is toting. YMMV.