One of the confusions here is that there are several different types of 'bindweed' and people tend to lump them all into one!
Field bindweed (Convolvulus arvensis ) forms creeping mats in arable and semi-arable areas (allotments!!!) and seems fine for rabbits (mine have been eating it for years).
Hedge bindweed (also known as 'Devils Guts' or 'wild Morning Glory'!) (Convolvulus sepium) seems also to be ok as long as not in seed.
True 'morning glory' which you often see growing in gardens as a blue coloured flower (Ipomea spp) is known as a mild halucinogenic:
However there is a suggestion that some other species/varieties of 'bindweed also have this halucinogenic quality in the seeds. .
SEE wiki QUOTE: The seeds of many species of morning glory contain ergot alkaloids such as the hallucinogenic ergonovine and ergine (LSA). Seeds of I. tricolor and T. corymbosa (syn. R. corymbosa) are used as psychedelics. The seeds can produce a similar effect to LSD when taken in the hundreds. Though the chemical LSA is illegal to possess in pure form, the seeds are found in many gardening stores; however, the seeds from commercial sources are often coated in some form of pesticide or methylmercury to retard spoilage [3] - these coatings are especially dangerous if you have a history of liver disorders and may also cause neurological damage. [4][5][6] Also, the active ingredient LSA has, in animal models, produced uterine contractions, which could possibly induce a miscarriage: END QUOTE
Note neither of those mentioned are Convolvulus . . the wild bindweed we have most commonly.
In conclusion I would have said leaves and shoots in moderation prior to flowering (and to be honest the same is true of plants like Cow Parsley - see other thread).