“Disabled for work” is not a natural phrase, no.
In the US and therefore with American English, we talk about being “on disability” or “receiving disability” meaning the public payments to workers who were able-bodied *until* they were injured/disabled on the job and became unable work anymore.
In American culture there is a very strong bias towards work. And there is no universal definition of “disability”. There is very little social recognition of people who were born disabled. To receive disability payments in the US, you really have to provide evidence of a work history, because the assumption is that disability is some condition you get as a worker.
And there is little support for, yes, that very common situation where disabled people can work somewhat, maybe part-time, but still need health benefits or some public assistance to live independently. There really isn’t a phrase for them.