Yes I think the vet has to actually see the lamb to assess whether the leg is broken or it has some other problem. If it's broken then of course the splint needs to stay on until it has healed. If it's a greenstick fracture then it will heal more quickly than a more adult type of break but I would think it needs to be splinted for at least a month. If you keep taking it off then any fracture cannot heal. If it is fractured then the leg will not be used normally without a splint, or it just might heal in a deformed position
Possibly the vet was using the usual criteria that a sheep is worthless so put it down if it's too much trouble and expensive to treat, not necessarily because it is not treatable, so explain that you can afford to treat and care for the lamb as it heals, but you need pain relief for it, usually something like Loxicom, expensive but effective
Let us know what the vet says