Rod Stewart was almost in Deep Purple? That's as weird as Les Claypool almost being in Metallica. I can't imagine Rod belting out Child in Time ...
They didn't think he was good enough after seeing him live (no joke!). The Jeff Beck Group was - along with Cream and early Led Zep - a pivotal influence on a lot of British bands. There might have also been a connection with Jon Lord as he was in a band/project with Ronnie Wood prior to Deep Purple (Santa Barbara Machine Head) and even prior to that had played with Ronnie's brother Art in the unsurprisingly named Artwoods. Anyway, The Faces and DP would stay friendly. DP broke America as the opening act of the Faces on their numerous tours, and DP never complained about their treatment and were full of praise for the backstage parties the Faces threw. Their only criticisam was, according to Ian Paice, that "the Faces were sometimes too knackered to play properly and that worked to our advantage as we would only party after our gigs", but that was of course part of the Faces' charm.
I doubt that Rod Stewart would have even seriously considered joining DP - coming from the Jeff Beck Group his work experience with guitar heroes had been such that he probably did not feel that it really warranted repetition. Instead he thought that the Jeff Beck Group bass blayer should play lead guitar in his new group.
Plus, and this is where your Child in Time argument comes in, Purple wanted someone with a Robert Plant type range to follow their original singer Rod Evans who was essentially a baritone. While the young Rod Stewart still had a great range, he wasn't into screaming above the music in the vein of Plant, Gillan, Byron or Ozzy.