Pretty sure that the game says to roll initiative when the DM says to roll initiative. Also pretty sure that there's sufficient slop in the surprise rules (like it not saying when surprise actually ends) that you can make a ruling that's consistent in a couple of different ways. That's not to say that you don't have a perfectly valid interpretation -- you do -- just that other interpretations can exist. Even those that don't agree with Crawford (blasphemy, I know, but it's a tolerate regime).
The relevant rules are (from the Players Basic):
Surprise
A band of adventurers sneaks up on a bandit camp,
springing from the trees to attack them. A gelatinous
cube glides down a dungeon passage, unnoticed by
the adventurers until the cube engulfs one of them. In
these situations, one side of the battle gains surprise
over the other.
The DM determines who might be surprised. If
neither side tries to be stealthy, they automatically notice
each other. Otherwise, the DM compares the Dexterity
(Stealth) checks of anyone hiding with the passive
Wisdom (Perception) score of each creature on the
opposing side. Any character or monster that doesn’t
notice a threat is surprised at the start of the encounter.
If you’re surprised, you can’t move or take an action
on your first turn of the combat, and you can’t take a
reaction until that turn ends. A member of a group can
be surprised even if the other members aren’t.
Initiative
Initiative determines the order of turns during combat.
When combat starts, every participant makes a
Dexterity check to determine their place in the initiative
order. The DM makes one roll for an entire group of
identical creatures, so each member of the group acts at
the same time.
Combat starts. Every one makes a dexterity check. Anyone who failed to notice a single threat at the time initiative is rolled is surprised (cannot take an action) when their initiative result comes up on turn 1.
Thats how it works. No free actions or free turns for people who attain surprise. Surprised creatrues simply miss turn 1 (but might still be able to take reactions if they are fast enough as reflected by the Dex check for initiative)