That is extremely creepy.

On the other hand,

• Obliviation does not require a hospital in any way
• Harry would probably snap and take Hogwarts apart brick by brick if the staff permanently took away his parents like that, and I don't think anyone involved is stupid enough not to get that

My interpretation is that Madam Pomfrey is a trustworthy staff member, while at the same time being inconsequential enough that her time can be spent on escorting Muggles around. She is also used to dealing with horrified parents, and therefore a good first point of contact for the Grangers.