14.33Half-diminished is different

Don't confuse that with:

Bm7♭5 → E7 → Am

Here:

Bm7♭5 = B D F A

This is the ii chord in A minor.

It functions as predominant.

Then E7 is dominant.

Then Am tonic.

So:

Bm7♭5 → E7 → Am

is:

iiø7 → V7 → i

Half-diminished is not the same job as G♯°7.