I've never heard of inserts going stiff after a strip like that. Did you do an extra rinse or two just in case it was that?
I think baking soda and bicarbonate of soda are the same, but baking POWDER is different - it's bicarb with some other ingredients for the baking process. Just use bicarb/baking soda if you want to try it, although I never have.
You could use soda crystals instead of the dishwasher tablet. This is the same stuff as bicarb but it is cheaper because it doesn't have to pass food safety tests etc as it is not a foodstuff. I use 4 x formula scoops to strip about 15 nappies.
My nappies or inserts are not hard when they come out of the machine after this, so dunno what's up there.
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