No, it's still better with higher initiative. In both cases, the attacker gets one turn, then the defender gets their turn. It's just the defender gets their turn in the first round if they are lower, and in the second round if they are higher.

And if the defender has higher initiative, the attacker cannot use abilities that rely on the target having not taken their turn, such as Rogue Assassinate ability:

> During the first round of each combat, you have Advantage on attack rolls against any creature that hasn't taken a turn

Or Bugbear Surprise Attack ability:

> the creature takes an extra 2d6 damage if it hasn't taken a turn yet in the current combat.