>>17115iirc amstrad's reason was literally just that the manufacturer was willing to give them a cheaper deal than any 3 1⁄2 disk manufacturer. (which backfired when they got stuck making unprofitable drives purely for amstrad since the alternative was a hefty payout for breach of contract)
like the famicom floppy itself, the reason it's a weird format existing in the first place to throw out low-ball bids to dodgy British people is that a bunch of companies wanted to design 5 ¼ floppy successors. iirc the FDS is from Misumi's attempt, Amstrad's is from Hitachi's, and the normal 3 1⁄2 standard is derived from Sony's.