Should primary and ESL teachers use an English alphabet that has the 44 or so phonemes that the language has?

“44 or so”.

And there’s your problem.

English phonology

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The vowels in Received Pronunciation group as:

(tɹæp) (bɑːθ pɑːm) (lɒt klɒθ) (θɔːt)

They group the same way in Australian English, though as

(tɹæp) (bɐːθ pɐːm) (lɔt klɔθ) (θoːt)

The vowels in General American, however, group as:

(tɹæp bæθ) (pɑːm lɑːt) (klɔːθ θɔːt)

You can do that, and teach the phonology of only one dialect—but at the expense of having to reteach them an intradialectal phonology later; in effect, you’d be reinventing English spelling.