Counterintuitive as it may seem, Archmage is not a spellcasting class. Instead, it advances another class’s spellcasting. So when practiced spellcaster says “pick a spellcasting class,” Archmage is not a valid choice.
You can see this discussed often online when someone thinks of applying Archmage or similar prestige class’s spellcasting advancements to mystic theurge (which is similarly not actually a spellcasting class).
WotC could have avoided much confusion had they drawn a clearer line on what classes actually were spellcasting classes. However, as a strong general rule, spellcasting classes have a class feature named “spells,” and the non-spellcasting classes have a class feature named “Spells per Day/Spells Known.” A few books instead use “Spellcasting” for both, which is less helpful, but the same approach applies: only classes that actually give you spells are spellcasting classes, the classes which advance other classes’ spellcasting are not.