Take this with a grain of salt, I heard it from the "color" commentator from a Bears game on TV but there are departures from the Tampa-2. In this case the Bears have a slightly more agressive version that blitzes more often than the conventional.
I have no evidence to support that, but why would a color guy lie to us?
As for an HC's background: Does it matter? A lot of coaches spend time on all 3 sides of the ball as they learn and craft their careers so it's not like the concept of offense is completely alien to some of these guys.
Also consider that in Indianpolis they brought in Dungy to turn around the D. 4 years later it's still an O show.
Marvin Lewis was supposed to be a defensive guru, but it's the Bengals' offense that keeps them in most games, when they were at their best on D it was because the other team had to pass a ton to catch up. Lately they've been ho hum.
Billick had the Ravens O swimming in the seas of lame before he finally took playcalling over (a long time into his tenure). The great D carried them.
etc, etc, etc
While a new head coach will try to lead with a blueprint he knows, his specialty means little to me as long as somehow, he gets results on the field.