Yes, mutual fund companies vote in shareholder elections.
They do, however, have a fiduciary duty to their clients who hold the shares of the funds:
Advisers are often charged with exercising voting authority on behalf of clients and must determine how to do so consistent with their fiduciary duties.
As a side note, "huge" may be an overstatement relative to the total number of shares. If you look at the largest AAPL Shareholders as of Dec 2024, no one fund company holder more than 9% of shares:
Could that be enough for one holder to swing a close election? Possibly, but they still have a fiduciary duty to the owners of units of the funds.