That's one of the reasons I sold mine.
My EB-0 and my SG (guitar) balance perfectly. Except for the EB-0L, my past ones did too.
60s EB0s/3s, with the much thinner/lighter body, don't dive per se, but balance in a very precarious way., like they"re ledge walkers contemplating suicide. Like a slight breeze and down they go. Moving the strap button from the neck joint to to top horn could take care of it, if you don't mind doing that. I have found 70s ones to be much heavier and not so, the shorties anyway.
I always feel better with a heavy axe - LPs are great that way; you know they're not gonna dive or move at all really if you let go to fiddle with something. My EB3 will almost float away.
Loving that Yammie, Morrow! There's been a few at Paul's lately, and I been tempted myself, but they weren't the ones I was most interested in (mostly pickup config).... and also it's really not in the budget/anywhere close to the top of the gear priority list. I did recently get an RNDI tho - those things are great.
S'up dudes.