Because of the way timing an priority works, your assertion that:
In most casual situations, it would basically come down to whether I
say "Hero's Downfall" first, or they say "+1" first.
is incorrect.
The active player always has priority on their turn until they pass it. This means they always have an opportunity to cast spells and activate abilities before their opponent(s) in each of the phases of their turn.
So the only time in which you have a chance to 'respond' is if they start the stack again or they pass.
Since the active player gets priority after the planeswalker resolves, there is no time you can play an instant before they activate one of the loyalty abilites, unless they start the stack somewhere else with another instant or sorcery.
116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives
priority afterward.
116.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player's mana pool, he or
she announces what mana is there. Then the next player in turn order
receives priority.
116.4. If all players pass in succession (that is, if all players pass without taking any actions in between passing), the spell or ability
on top of the stack resolves or, if the stack is empty, the phase or
step ends.
So here is how it breaks down:
- Player A puts a planeswalker on the stack.
- Noone wants to add to the stack, so it resolves and enters the battlefield as a permanent.
- Because the Active player now has priority again, they have the opportunity, as a sorcery, to activate a loyalty ability of the planeswalker and they do.
- Player B casts Hero's Downfall in response.
- Everyone passes so Hero's downfall resolves. The planeswalker is destroyed.
- The planeswalker's loyalty ability resolves because it is it's own entity once it is in the stack.
602.2a The player announces that he or she is activating the ability. If an activated ability is being activated from a hidden zone, the
card that has that ability is revealed. That ability is created on the
stack as an object that's not a card. It becomes the topmost object on
the stack. It has the text of the ability that created it, and no
other characteristics. Its controller is the player who activated the
ability. The ability remains on the stack until it's countered, it
resolves, or an effect moves it elsewhere.
Another way it could happen:
- Player A puts a planeswalker on the stack.
- Everyone passes, so it resolves and enters the battlefield as a permanent.
- Player A activates the ability of Gilder Bairn.
- Player B cast's Hero's Downfall.
- Everyone passes.
- Hero's Downfall resolves and the planeswalker is destroyed.
Because the loyalty ability of the planeswalker is not an instant, but a sorcery, they can't activate it in response to Hero's downfall and the planeswalker dies before the ability can be used.
It comes down to loyalty abilities can not be activated in response to anything because they are cast as a sorcery, not an instant.