Presumably mně is allowed instead of mi. Is one preferred over the other in modern speech or writing?
Nope. Those two are completely equal. In similar manner you can use mě and mne (without háček) interchangebly
It also works for second person informal TI and TOBĚ for dative.(and TEBE a TĚ for acusative]
To a certain extent I agree with you, but I wouldn't use "Pojď ke mi" instead of "Pojď ke mně" (Come to/towards me).