Does 慢一点 mean "be careful" even in a general context? I've been told by a native speaker that 慢一点 is functionally the same as 小心一点, which in english is essentially "be careful". Can someone else confirm?

I find it a little odd and would consider this a mistake if it was in english, especially if being slower doesn't help (e.g. you're already doing it slowly but carelessly, or you need to do something quickly but decisively)
I don't think that 慢一点 is the same as 小心一点. 

I am from Manchuria, and as far as I know, sometimes when people are gonna leave each other, a person may say 慢走 or 慢点走 as an expression for "Goodbye". Maybe a personal bias, I don't think 慢走 is frequently used. 

I also agree with you that, doing things slowly may not always help.