The initiative is sort of an illusory concept: you won't be able to find a
Law of Poker entry stating "The player who put in the largest raise pre is obligated to take the betting lead, and any violators will be imprisoned immediately".
Usually whomever has an advantage in some combination of equity (both showdown and ex-SD) and polarity (i.e., "nut / near-nut hands", the root word being "polarization" -- this one usually matters more than equity) should take the betting lead.
Imagine you were playing HUNL with weird stipulations where the BTN had to open 2-3x (or whatever) with any two suited cards and any pocket pair and fold everything else and the BB had to flat with [8-A][8-A] (any two cards eight and up) and fold everything else. The BB would gleefully start donking on flops like AK9r, but wouldn't dare on 542tt.
In HUNL in SRPs, the BB is usually around equal or worse on most flops wrt polarity, and being IP confers the BB an ex-sd equity edge so it's not like the BB is exhilarated over having a 52-48 showdown equity edge or anything, and balancing a donking and checking range in ~symmetrical situations is a hell lot of work for very tenuous/marginal gain. So donking on the flop isn't really a "thing" yet, except exploitatively against fish I guess. On the turn, however . . .