I agree with you that it's too early to think about who the next permanent HC is and that offering Rizzi the job right now is a bad idea, we have an opportunity to see how he performs in this role as well as how the players respond to his coaching in terms of effort, camaraderie, even the kind of things they put out on social media for that matter. The organization should take full advantage of that.
I'm not sure who's going to be in charge of hiring at this point (though right now it would be Loomis) but from my perspective the promotion of Dennis Allen was under such odd circumstances that I'm willing to give the front office a bit of a pass on it. They clearly didn't believe Sean Payton would actually walk away (neither did a majority of the players), if I remember correctly Gayle even mentioned during Payton's retirement presser or the frenzy around it that she actually floated the idea of a one year sabbatical which Payton declined. The group of players we had here, even without the HOF QB, was still relatively young/entering their prime only 5 years removed from that 2017 draft. There was hope that Michael Thomas could be Michael Thomas again, defense was looking like a potential juggernaut (minus the end of halves), etc. On top of all that by all accounts Payton himself recommended Dennis Allen. Allen had just rebuilt our defense from horrible to capable of making a Super Bowl (everyone knows what happened in that NFCCG). His defensive players at least seemed to have his back. The only thing that pointed against it was his time in Oakland where coaches have been failing pretty much since Bill Callahan's Super Bowl appearance. Turns out that red flag should've been a black flag but in my opinion them giving Allen a shot wasn't the worst decision at the time. With the benefit of hindsight, it obviously was a bad one.