Edit: BTW Today, 10th May is Maid Day in Japan. Coincidence or intentionally uploading this chapter today?
I think it's more a case of no one expecting the person in front of them to have dyed their hair. Most people won't look at Melody or Anna and think "maybe she is a noble lady or a saint in disguise". Lect only realizes Melody is Celesti because his previous job was to look for Celesti and by coincidence, he finds out Melody can change her hair colour easily. Later a reincarnator also recognized Melody but she is a special case
BTW this was lampshaded in the LN, when Melody saw a modern style bathroom, she thought it was convenient.
Because at that point they were desperate. Since some of the key events still happened without the heroine (or so they thought), they knew they needed the saint's power to defeat the demon lord, so they are hoping they could find the heroine during one of these events. In fact, this date only reinforces Annemarie's theory that the world is self-correcting by substituting other characters. (This will continue to be proven true, but I'll leave the spoilers for later)
Poor Sir Lect, even the readers forgot to include him.
Also next is the orphanage visit!
... this wasn't covered in the LN, I think? Curious how this will play out. This is probably the same orphanage that will have minor importance in the next arc