Good study. Backs up my own anecdotal observations. In keeping Incilius alvarius toads for over ten years I am, however, totally sold on vitamin supplements.
Currently I rotate between Zoo-Med, Nekton-Rep and Herptivite. The first two have preformed vitamin A and D and the last beta-carotene and no D. I believe this gives them lots of A and a break from A & D (which can build up) when they get the Herptivite. I dust heavily at every feeding mixing the vitamins about half and half with calcium with NO D3. I feed my toads in a separate container so they eat them all quick while there is still lots of dust on them. My toads seem really healthy.
I kept and bred dozens of snakes for over a decade when I was really into them and they always got Nekton-Rep in their water too. My first toad went it's first 10 years with only Nekton-Rep dustings but I read a zoo-keepers advice to rotate a few different ones to negate the effect of a weak batch etc because the quality control isn't the same as with human supplements.
My first toad went 10 years on crickets with the occasional nightcrawler and monthly an appropriate sized mouse. I think the mice helped with calcium perhaps and there is some calcium in Nekton-Rep too.
When I got my newest two toads a year ago they couldn't "target" very well with their tongues. Aggressive dusting regime fixed that pronto and they are super-healthy now. I think it was indeed about the vitamin A.
For the record, my toads swim around in tap-water that has been through a Brita filter. 10 years of bottom-of-the-pool-toads says they don't mind soaking in it. This was in two different cities. Your mileage may vary but I don't think chlorinated tap-water is always as toxic as people make out or maybe the Brita gets the job done. I like it better than distilled because I think that might leach minerals from animals.
Leaving the water sit over-night will get it about as chlorine free as Reptisafe (at 24 hours sit there will be NO chlorine). I don't like adding chemicals to anything an amphibian sits in. Brita works for me but a 24 hour sit would mean no chlorine if you want to be safe.
I hope things get better. Here are some quick pics of sleeping (? I think they sleep. Maybe like whales and dolphins.) and woken-up Rx and Mojo (on left, last pic). Two snuggly toad brothers not currently in their heated swimming pool.