HI Daniel, thanks for your question. In over 35 years of working in horticulture, this is the first time I have ever heard of mulch glue.
In Extension, we are charged with providing you research based information. In researching at our University Library system, I can find zero references to actual research on these products. In addition, the articles I find on this (that aren't thinly disguised advertisements) are very vague, saying things like the glues "should be environmentally safe" . I can't find any data anywhere. I did pull up a SDS (safety data sheet) on one product:
https://cdn.sanity.io/files/lxi9ex2x/production/88cd275e02ab5087a7c9176244c73e5706f50a98.pdf
I just chose this one at random. It looks like there are toxicity issues with this product with skin and oral entry routes. No environmental data was available.
I found a couple references to water issues with plants that are not well developed- they can't access water well if the glue is applied before the plants are established. That raised some concerns in my mind, but again, no data.
Looks like that was an awfully long winded way to say I don't know! Thanks for bringing these glue products to my attention. For now, you'll have to study the SDS sheets on the various products and decide for yourself if they are right for you and your situation. If I am able to garner any more info from my network of horticulture pros, I will share it with you here.
Many thanks, Nicole