Proficiency and spell attack are correct, but you use NPC tools, not PC to verify.
You are correct in that a 9th level PC spellcaster should have a proficiency of +4.
But the Mage NPC (MM, 164) has a +3 proficiency.
If we were calculating as a PC.
But this is a NPC and we use the NPC rules around proficiency modifiers.is most likely a result of the proficiency modifier differences between PCs and NPCs.
Page 8 of the Monster Manual covers the Proficiency Bonus by CR and it lists a CR6 Monster with a bonus of +3. It isn't until CR9 that they achieve the +4.
So while the Mage is listed a 9th level caster, their proficiency is based off of that table and is +3.