I think it's a mixed thing. I know many of our members like them as they reply to them to say thanks. On the other hand I personally don't care about them and would like to disable them as I know they are automated and so there isn't actually someone caring about my birthday on forums where I am registered. This is a good example of a feature that needs the on/off switch in the member's control panel.
I agree that this whole area has pros and cons.
First I think a generic display on the forum is much more acceptable as an auto message than an email. Notices, or blocks of sidebar info, or stats at the foot, are all recognisable as auto info.
One option might be a Today's Birthdays link, with a further option on its page for members to send all those members an HB message - with the member adding a custom message. Possibly a privilege by usergroup to do it to cut spamming. Admin requires custom message of 50 characters to deter spamming?
Wouldn't suit all boards but could be fun.
Secondly for those who don't want an HB email this needs a Pref disable.
For me although I know it's automated I still like getting an HB email. On that one day clutter in my inbox is warm n fuzzy. All my contacts should celebrate ME on that day!
There is a personal element though and could be more. The personal element is that the admin bothered to enable the service and added their own message. Mine for example when I used another forum software was an old traditional Celtic blessing which people loved. (May the road rise up ...)
This suggests that customising the birthday message to fit type of forum is important, not just a generic HB, and
perhaps further customisation by usergroup as an admin option.
Third option to show a birthday Notice to the member is great. With graphics n stuff. Poss customised to usergroup.
Only they have to visit to see it so why not make the email tell them it's waiting for them?
Plus I think an option to see it for a 3 day period would be good to help reclaim inactive members. Birthday celebrations and visits easily extend over several days, or even a week. So, finally, a 24 hours reminder if member does not visit gives a double whammy on the reclaiming inactives.
This could be a power Birthday addon.
A variation could do something similar on the anniversary of joining the board. Different options for active and inactive users.