Why not use the 4 digit pin and then loop, appending the first two digits to the end to get a 6 digit pin? It’s somewhat easily guessed, but is a hell of a lot better than 333333...
It also have the advantage of not having to save it in a password manager, since I have it on my calender (with a lot of other birthdays saved).
4-digit Birthday-pins are probably good as long as they're not your birthday or that of your kids, and you do the last 2 digits of year, last digit of month, last digit of day, go capture as much of the entropy from the date as possible.
On a side note, I remember back in 2004, a colleague born on Feb 29th, 1984 was unable to enter the U.S. H1B renewal website because someone forgot about leap years in their date validation logic.