No, spontaneous spellcasting archetypes cannot swap spells
The ability to swap spells is not part of the Spellcasting feature for the class that the archetype emulates, it is part of the Spell Repertoire feature.
Let's look at an archetype to see how it works. The sorcerer archetype (as an example for a sponteanous spellcasting archetype) says
You cast spells like a sorcerer. You gain access to the Cast a Spell activity. You gain a spell repertoire with two common cantrips from the spell list associated with your bloodline, from the spells granted by your bloodline, or any other cantrips of that tradition you learn or discover.
The archetype says it casts spells like a sorcerer, so it gains access to the sorcerer's Sorcerer Spellcasting feature. It however does not say that it learns spells like a sorcerer, so you do not gain access to the sorcerer's Spell Reportoire feature that includes the ability to swap. Instead it explicitly describes the nature of the spell repertoire you gain.
This is also not changed by the various spellcasting feats. The Basic Sorcerer Spellcasting says
You gain the basic spellcasting benefits. Each time you gain a spell slot of a new level from the sorcerer archetype, add a spell of the appropriate spell level to your repertoire
and the Basic spellcasting benefit for your repertoire is
these feats grant a 1st-level spell slot. At 6th level, they grant you a 2nd-level spell slot, and if you have a spell repertoire, you can select one spell from your repertoire as a signature spell. At 8th level, they grant you a 3rd-level spell slot
so for each of these spell slots you get, you may add one spell of appropriate level to your repertoire. Expert and Master spellcasting for the sorcerer just refer back to the standard Expert and Master spellcasting to expand this, for Expert:
grant you a 4th-level spell slot. If you have a spell repertoire, you can select a second spell from your repertoire as a signature spell. At 14th level, they grant you a 5th-level spell slot, and at 16th level, they grant you a 6th-level spell slot.
And for Master:
grant you a 7th-level spell slot. If you have a spell repertoire, you can select a third spell from your repertoire as a signature spell. At 20th level, they grant you an 8th-level spell slot.
None of them mention anything about swapping spell selection, so you cannot swap.
The Learn a Spell action refers to a situation where you "add or swap spells". You can apply it whenever you get to add a spell to your spell repertoire. As you are not swapping spells, you cannot apply it to swapping spells.
Retraining the feat
Everyone however can retrain feats (kudos to András), if the DM provides for a teacher. The rule for retraining feats states
You can spend a week of downtime retraining to swap out one of your feats. Remove the old feat and replace it with another of the same type. For example, you could swap a skill feat for another skill feat, but not for a wizard feat.
As you select your spells each time you gain a spell slot, and you would gain the spell slots anew, you could then select new spells for them. As you have to select another feat, you may not be able to swap one of them for itself by retraining, but you could spend two weeks to first swap it for another Archetype feat and then retrain the spellcasting feat.
If you have access to the higher level expert and master feats, and wanted to change a basic spell, this would be even more cumbersome, as you would need to retrain away and back in the whole chain (they are dependent on each other), but it would technically be possible.