Letters are about Style:
I frequently use postscripts in my stack exchange answers when I have added an element in editing to an existing answer. If, for example, I reread someone's answer and realize mine partly overlaps, I want to acknowledge the other person's answer. I also want to keep the original content as it was while adding some piece of information to the answer, and a postscript draws attention to the addition afterwards (especially if someone rereads my answer and sees a bullet-point addition at the end.
Today, writing letters is a very old-fashioned kind of thing to do. It's a statement of a well-thought-out and personal address to another person, and intended to be a little more enduring than a generic email. As such, a postscript is a charming element to a letter. You admitted that the whole point of the letter is really an excuse to add a clever postscript. I like to structure personal emails in a letter format, and postscripts are a good way to do this.
It's especially appropriate if the detail in the postscript is different from the rest of the letter. It can and often did draw attention to a small detail you wish to address, less informative and more intimate. It was, admittedly, originally intended to meet the need for appending a detail to a letter that was not editable. But the reasons for a postscript as a style choice are as diverse as readers and writers.
Hand-written letters are the most intimate kind of letter of all. It is again a style choice if you want to use a different writing tool (or font, if printed/emailed) and people are not going to know your motives in writing a postscript so I doubt it could be construed as rude or offensive. Indeed, due to the highly personal nature of a letter, you'd know better than anyone else if the intended recipient would take offense at it. If they do, then send them a text. I think it would seem a bit weird to send a postscript as a new letter, so I wouldn't recommend it. There are lots of media nowadays for communicating a small snippet of information.
So while postscripts are admittedly old-fashioned and technically outdated, I find them pleasant and charming in a format that often lacks character today. Feel free to do what feels right in a deeply personal format that's all about personal contact and style.
- PS. I love a good bourbon, especially in a nice old fashion. Scotch just recycles bourbon barrels. ;)