I don't think Akechi inherited anything from Shido except for one personality trait, which Sojiro sums up on 11/23: "[Shido] can't sit still until he crushes whoever opposes him, no matter how trivial."

The difference is Shido's idea of opposition is endless. Akechi's isn't.

Akechi and Shido both share relentless determination, but where Shido's is based solely on selfish desire and pompous malice, Akechi's is rooted in two different things: vengeance, and competitive inspiration.

Akechi is relentless both in his pursuit of taking Shido down, and in his rivalmance with Joker. His relentlessness *can* lead to problems (see also: vengeance quest), but as a trait itself, indefatigable determination is far from a bad thing. Joker's that way, too.

This doesn't make them sore losers; they're boys who push themselves to the limit to achieve their ambition. When it comes to their relationship, Akechi doesn't want to not be Joker's equal, nor does he want Joker to change from being HIS equal.

So, what am I getting at here?

When applied to something healthy, Akechi's relentlessness becomes a focus, almost to the point of devotion, to the object of his affection. We can see this with how much he still loves and honors his mother's memory. Joker is the next example.

He also applies himself in this way to his studies, his work as a detective (however much of a sham that is in game [the anime showed differently!]), and his public facade. The boy gives 100%, or nothing at all. He can't be any other way.

And neither can Joker :)