I had a go-round with MF on Twitter on this very topic this week. Daly is not injured; he has osteoarthritis, as defined by the ADA. This is not up for debate, whether "you" consider it an injury, a syndrome, a condition. The European Tour does not have an ADA-equivalent proviso for carts, and this would be quite challenging to effect, given the number of Yes-Union and No-Union host countries along its tour schedule. Quite separate from the tour, The Open Championship also does not have such, and is under no decree to offer a ride to JD.
The R & A did a proper thing in denying the request. It played the hand of "govern us to do it and then we'll consider it." Once it allows a non-legal precedent to be set, it is open for myriad challenges and requests. The day will come, with The Open as with all golf events, when such a court case is opened.