Doesn't the surface area increase as the BH takes on more mass?

I like the analogy of a snowball, nothing ever enters it yet it grows.

This accurately describes how I gain weight as well.

This joke is in direct violation of the First Law of Thermodynamics.

Yes, but the mass does not have to cross the event horizon for that, does it?

At that point we probably can't tell more without diving into the math, and it's beyond my abilities. Yet the gross idea does not seem absurd to me.

Yes, the radius of its event horizon is directly proportional to its mass. The relationship is described by the Swartzchild equation.

What is important to remember is thst our point of view is not special or “right” in any way. There is a difference between our observation of a process like matter falling into a black hole, and the reality of it. Black holes do “eat” and grow, and that occurs despite the intense time dilation near their surface. The light which returns to us will be redshifted to black, and we won’t see the final event, but it does happen in its own proper time, in finite time. When black holes merge we can detect their gravitational waves, evidence that black holes are not “frozen” in reality.

It is also important to distinguish between the implications of a model black hole formed from the uniform collapse of a perfectly spherical dust cloud, from infinity, in an otherwise empty universe, with no charge or angular momentum, from what happens in nature.