Do the French really not have that problem though? It seems to me that most people know the difference between "your" and "you're", they just don't care enough to use the right variant.
Regarding your question about French, I'm not too sure. There/their/they're would be la/leur/ils sont, so perhaps there's scope for the first two to become conflated, not so much the last.
As far as I know the biggest issue French is facing is the injection of English words where there isn't a French equivalent.
Then again, French people seem to absolutely loathe the version of French spoken in Quebec so there could be a plethora of other issues involved that we don't have in English.
Regarding your second point, personally I'm not sure but ignorance certainly plays its part one way or another.
On the one hand I agree with you. The English we use now, even when written or spoken in a grammatically perfect way, is very different to the English used a couple of hundred years ago.
On the other hand, seeing "your/you're" or "they're/their/there" used incorrectly or interchangeably rustles my jimmies immeasurably.