It's not overpowered, this is just its niche.
First, keep spell level in mind. Animate Objects is a 5th-level spell. Comparing it directly to counterspell, dispel magic, or fireball, all 3rd-level spells, is already a lopsided comparison (and the specific spells make it worse, but I'll cover that later).
Spells are also extremely situational. Even looking purely at those designed for combat, which one does best depends on many factors - number of foes, positioning, expected duration of combat, freedom to concentrate, etc.
In this instance, the objects have only one target, and can direct their full 10d4+40 damage at them. Assuming they all hit, that's average 65 damage. It seems like a lot, but consider these:
- That's to only one target. Area of effect spells like Cloudkill or Insect Plague (also 5th-level) get better with more targets in a way that Animate Objects doesn't (to deal damage to multiple targets, the objects must split their efforts, dealing less damage than if they ganged up on one). Cloudkill, for example, may only deal 5d8 (average 22 damage), but to each target in a 20 ft radius: each additional target takes all of the damage. Even at only three targets, it's already dealing slightly more damage than Animate Objects.
- It can be interrupted. Animate Objects and Cloudkill may deal their damage every round, but if you lose concentration, if targets move out of the way, or if the objects get 'killed', that may only be once. Cone of Cold, also a 5th-level spell, deals its damage only once, but deals much more: 8d8, average 36, again per target. Even with as few as two targets (and its 60-ft area is huge), it outdamages Animate Objects for a round. With four, it takes Animate Objects two rounds to overtake. If your foes have few enough HP, dealing damage up-front can be more important than dealing a lot of it.
Finally, the spells you bring up that your mage doesn't have are very common for spellcasters to have - so much that this mage not having them isn't really a fair comparison, and does more to highlight the weaknesses of Animate Objects than the strengths.
Put in a different situation, Animate Objects won't be as powerful as it is here.