It doesn’t “reset” anything, but rules as written, you can make an extra attack as part of each Attack action taken that turn.
The wording of the ability is (emphasis mine):
If you take the Attack action on that turn, you can make one additional weapon attack as part of that action.
This doesn’t include any restrictions on how many times the feature may be used, only when it may used (“on that turn”, i.e. the first turn). This is in contrast with features like the Rogue’s Sneak Attack or the feat Savage Attacker, both of which begin “Once per turn”.
It’s also clear (with thanks to kanoo in the comments) that it is specifically the Attack action that gains the extra attack, since it is “part of that action”. So if you use Action Surge to take two Attack actions on your first turn, each Attack action meets the requirements and gains the Dread Ambusher extra attack.
There are very few ways to get additional actions - aside from Action Surge, the main one is the haste spell. Haste has limitations which rule this out: using its extra action for the Attack action is specifically limited to “one attack only”. That is not a universal rule for extra actions, though, and mostly necessary because haste lasts multiple turns. So I think this clever class features combo works under rules as intended (and certainly rules as fun), and will be fine at most tables.
As a comparison, this combo doesn’t grant access to extra attacks any faster than a single-classed fighter or ranger, since it requires ranger 3/fighter 2. A 5th level single-classed fighter has Extra Attack every turn and can use Action Surge whenever they like during the fight to get four attacks in one turn. The multi-classed Gloom Stalker can only do this on the first turn of combat, though they also get the speed boost.
After two or three more levels (depending if they progress as a fighter or ranger) the multi-classed character will gain Extra Attack and can thus make up to six attacks in that first round by 7th or 8th level - a few levels earlier than a fighter gets their upgrade to Extra Attack at level 11. But they’ve paid for it by delaying ability score bonuses, spellcasting progression and other features, so this seems pretty reasonable to me.