latepaul, notice that in your first example "a person walking on ice should take care so that they don't slip" does not read as third person singular, it uses "they don't" rather than "they doesn't".
I leave it up to someone with better formal education in this area to offer the right words of technical explanation, but what I do know is that translating "he doesn't" into "they don't" is something that causes me to stumble while reading.
* George should take care so that he doesn't slip. (Reads okay)
* George should take care so that they doesn't slip. (Sounds like Gollum!)
* George should take care so that they don't slip. (Stumble: Huh? George and who?)
That dictionary.com article would seem to suggest that Gollum got it right (since that article doesn't discuss verb agreement), but the third example is what I've seen used in practice.
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