I think
However, doing so is different from "interacting", which (it turns out) is a term with a definition, that normally takes an action.
That's not quite correct. in the OP you say
The PHB tells us (p. 24):
Interacting with Things. You can interact with one object or feature of the environment for free, during either your move or action. For example, you could open a door during your move as you stride toward a foe.
Meaning "interaction" is a non-action that occurs as
part of another action/move. So you could poke a suspected illusion on your turn for free, assuming it was stationary. And that its an illusion of an "object or feature of the environment". That rule wouldn't seem to apply to creatures so players wouldn't think to try.
So back to the other RAW, the wording in Minor Illusion is "...Physical interaction with the image reveals it to be an illusion, since things can pass through it."
Any physical weapon applied to an illusion would be "things pass(ing) through it." In melee, any stationary image larger than, say, a coin, should be an automatic task, not requiring an attack, even if the only things in your hands are weapons.
However, based on having seen people miss nails with hammers, making an attack roll and using the attack action seems reasonable for hitting very small objects.
Ranged interactions would require using an attack action as "throw things at another thing" is the definition of a ranged attack. It may not require a roll, as a stationary illusion of a goblin at 10ft is a gimme, but the same illusion 50ft away should require a roll, given that a lot of weapons are at long range/disadvantage.
the very prominent statement about rolling an Intelligence (Investigation) check is functionally meaningless, because it sets up a challenge that can be failed when tossing a copper piece or making a ranged attack would automatically succeed
That's not true. Physical interaction doesn't invalidate "study" as a use case, as there are times you don't want to walk up and poke something. "Are the Crown Jewels real? Guess I'll just go give'em a poke or throw stuff at them." Nope, doesn't sound good.
More common scenarios are hidden/invisible characters who don't want to risk being detected or where the suspected illusion is implausible to attack, perhaps behind a wall of force.
So rules on identifying illusions without any physical interaction are needed, even if other options exist.
But what the rules don't cover are animated illusions, like Major Image, where a caster can spend their action causing the illusion to act/react. How do you poke a suspected goblin illusion that jumps around wildly, staying at least 10ft away? Or it swings its axe wildly, but doesn't hit your shield or sword?
Or illusions of intangible things. You throw a coin at a Ghost and it passes through....how would that tell you it's an illusion? That would happen normally.
Those are gaps in the RAW