"The Continental United States" should include Alaska since it's still a part of the same continent, but common usage of the phrase seems to exclude it anyway.

It does...

What you're thinking of is "contiguous".

Continental: everything but Hawaii, 49/50

Contiguous: everything except Hawaii and Alaska, 48/50

The phrases "Lower 48" or "contiguous 48 states", are more typical than "the continental United States".

The related but distinct term continental United States includes Alaska, which is also on North America, but separated from the 48 states by British Columbia in Canada, but excludes Hawaii and all the insular areas in the Caribbean and the Pacific.

That's because continental USA includes Alaska. You're thinking of contiguous or conterminous USA.

I've heard it better described as the "Contiguous United States", which covers the fact that you can't get into Alaska by land without exiting the US.