You can walk home if you get your crutch. (Can should be replaced with “will be able to” otherwise it sounds like the other person will be permitted to walk him if he gets a crutch)
This is the disagreement. There isn't actually any need to change this sentence. "Can" is often used both for "able to" uses and "permission to" uses:
- If you finish your vegetables, you can have dessert.
- If you look through the telescope, you can see Jupiter.
Context and/or tone help us tell the difference. Eating dessert often requires permission, while seeing Jupiter doesn't. The first example is probably saying "I will let you have dessert only if you eat your vegetables," while the second is probably saying "You will be able to see Jupiter only if you use a telescope." One could maybe invent unlikely scenarios that would have the opposite meaning ("Hey, you can have dessert any time you want, but look, there's no room on your plate! The vegetables are in the way"), but if we were in this unlikely situation, we would know it.
As is often the case, if we have a single isolated sentence without its real context, we can't be 100% sure of its meaning. But the example about walking home with a crutch is more similar to the telescope/Jupiter example than the vegetables/dessert. A crutch is a tool needed for walking. There are few situations in which someone is held against their will and prevented from going home, and of those situations, it's hard to imagine one in which getting a crutch changes the permissions.
Note, as mentioned in the comments, if permission is intended, then "may" can be used with no ambiguity. Also: you're right that "abilities you can gain" is important. "Can" can only be used for an ability you have or can actually gain. If it's something you can't do and can't become able to do, you can't use "can" and should use "could" or something similar: "If you had wings, you could fly."
(To those who would insist that you must not use "can" for "permission" uses, Merriam Webster says:
Can and may are most frequently interchangeable in uses denoting possibility; because the possibility of one's doing something may depend on another's acquiescence, they have also become interchangeable in the sense denoting permission. The use of can to ask or grant permission has been common since the 19th century and is well established, although some commentators feel may is more appropriate in formal contexts.