[RPG] Does breaking a magic item render it useless

Generally, no...

The Adventurers League Players Pack contains the documents that detail your options in AL play. Purchases are restricted to items in the PHB. The only potion available for sale without special documentation is the basic Potion of Healing. There are no other magic items in the PHB - no potions, no scrolls - so they can't be purchased either.

...unless you're of sufficient rank in a faction...

Characters who reach rank 3 in their faction can use downtime and gold to acquire magic items from a very limited list: +1 weapons, shields, and armor are available to all faction members, and each faction allows access to a single uncommon item and a single rare item.

...attend an event featuring Fai Chen's Fantastic Faire...

Fai Chen's is available to public events of sufficient size that register with WoTC for it in advance. Fai Chen trades (but does not sell) permanent magic items, sells scrolls up to 5th level (one per transaction per event), strange trinkets, and occasionally special mounts and property deeds.

...or adventure in Chult.

There is a very small number of magical weapons and shields, plus scrolls and potions, available for purchase only to characters playing Season 7 content - DDAL modules and Tomb of Annihilation. These purchases are only available as a downtime activity.

The Arcane Savant can do this.

The Arcane Savant is a prestige class that focuses on getting the most from magic items.

At 3rd level, they can use their caster level on spells they cast using scrolls and wands.

Scroll Master (Su)

At 3rd level, an arcane savant can use his own caster level instead of the item’s caster level when using a scroll or other spell-completion item.

Since the Arcane Savant prestige class requires 5 hit dice as part of its prerequisites, the earliest a character can gain the Scroll Master ability is at 8 hit dice. This is faster than the Magician bard archetype, who gains a similar benefit at 10 (or more) hit dice.

At 10th level, the Arcane Savant gains their capstone ability, and can use their caster level for any single magic item.

Item Master (Su)

At 10th level, an arcane savant can spend an hour focusing his energies on a single non-charged magic item that requires an action to activate. At the end of this hour, that item becomes attuned to the arcane savant, and from that point on, that specific item can use the arcane savant’s caster level to resolve its effects rather than the item’s caster level. An arcane savant can change which item is attuned to him by repeating the attunement ritual, but he loses attunement to the previous item. An arcane savant can have only one item attuned to him at a time.

Note that the Arcane Savant is behind one caster level, although you can use the Magical Knack trait to compensate. So if you multiclass Wizard 5 / Arcane Savant 10, then you could use a Mask of the Skull's effects at 15th level, as you described.