Äh?

From time to time, I come across English words being adopted into German with the A rendered as an Ä, for example Känguru. I don't get it. As I understand it, short A in English represents the sound /æ/, which sounds, to me at least, far closer to the German A (/a/) than Ä which, as a short vowel, represents /ɛ/, the sound in English "bed". (/a/ is apparently the first half of the diphthong in English "ice".)

I'd put this down to Germans having a tin ear for the subtleties of English vowels (in much the same way I do the other way around) until Sunday, when Rafi, who has a good grasp of the sounds of both languages (but, as the remainder of this sentence will show, a poorer one of German orthography, which is fair enough given that he hasn't started school yet), announced he was going to write "Daddy" with German spelling, and wrote "DÄDI" in pavement chalk.

Can anyone unriddle this mystery for me?