In general it seems awkward, imprecise and slightly infantile-sounding. To me. But that may be my English ear (I have one of each!); it’s also a transatlantic issue. In the USA people routinely use “that” to mean “who”; I did it myself and didn’t hear myself doing it. You even see it in places like the New Yorker or the NYT. But even in the US I think it’s essentially a colloquialism and doesn’t work very well in print.

I’m one that does this/ I’m the one who does this. Is that it?

It’s also handy to avoid repeating ‘who’.

I almost always use ‘who’, but I do believe ‘that’ is sometimes acceptable or even better.

I think it’s more specific than just when referring to a person – you wouldn’t use ‘that’ when referring to a name, but you would with a normal noun – so ‘the fellow that can barely string a sentence together’, but ‘John Prescott, who can barely string a sentence together’.

There’s a handful of clever relevant quotes about being excessively fussy, by chaps like Samuel Johnson, Mark Twain and GB Shaw, that I could trot out here. Except I can’t remember any of them right now…