1. He found the spell list uninspiring (admittedly he was only looking at the core book and not the companion, which I've now directed him to).
The spell list in the main book is an artifact of the game ACKS is derived from: B/X. Perhaps the expanded list in the Player's Companion will help. However, I can't really recall any Basic D&D games where anyone ever complained about the selection of spells. I'd be interested to know a bit more about specific objections...I wil say that one of the tenets of the style of play generally associated with ACKS (i.e. old-school) is of coming up with creative uses for those spells. That means it can be important to think of uses beyond the straight descriptions of each spell, in how they can be usefully applied.
2. Changing spells seems like really hard work, particularly with respect to cost.
Based on the designer's notes about this, it's intended to be a kind of halfway point between the D&D 3.5 Sorcerer, and freely swapping spells (which would render the concept of a Repertoire largely redundant). My experience in actual play is that it works quite well, in spite of how you might think it reads. Within a level or two, the amount of gold necessary for swapping a spell is not such a big deal, and the Mage will often have the time while more martial classes are recovering from Mortal Wounds. Also, as I touch on in my reponse to the Spellsword question, Mages will tend to have a larger Repertoire which renders this less of a pressure in actual play. The fact that there are fewer spells which directly solve specific problems, and that the acquisition of them is largely random, also alleviates the pressure to swap to "that perfect spell."
3. Why wouldn't one just play a Spellsword, who seem to be "strictly better versions"?
One certainly can, and for a player thinking in those terms, I would probably suggest they should.
There are some other, more subtle interactions beyond the difference in straight XP requirements, though...
First off, Spellswords have two Prime Requisites, and.as a martial class, there is also more pressure for them to have decent scores in other Attributes beyond those (e.g. Dexterity for Initiative, AC, and missile attacks, and Constitution for Hit Points). This means it's quite common for a Mage to have a much higher Intelligence, and thereby receive Bonus XP (+5-10%), meaning a Mage will likely level even faster than the Level Progression table would imply, and will have a larger Repertoire and more bonus Proficiencies.
Spellswords will often be carrying weapon(s) and possibly a shield, meaning their hands are full, and they cannot cast. A Mage will rarely be carrying anything they can't drop in order to cast a spell.
Spellswords are often going to be putting themselves in harm's way, with their higher Hit Points, AC, and weapons. This tends to make them less survivable, in spite of the improved HP, AC, and weapons suggesting the opposite, as they will simply be facing more than enough attacks to overwhelm the improved probabilities from the aforementioned scores.
Ultimately, Mages play quite differently than Spellswords, and that difference in playstyle is (for me) the reason to play one over the other.