Investigation check:
Hm...but that would somehow totally negate the "The target rationalizes any illogical outcomes from interacting with the phantasm" part of the spell's description. I believe, WotC would just have written that the investigation check would (or could) be triggered after the target has been rationalized inconsistency once. But they didn't which I gather from that it's not intended to end the spell effect so fast. As a player, I would be a bit disappointed, if my DM would end that spell effect so fast.
Yes, but you know that if interacting with an illusion does trigger rationalization, how do you investigate it? That's the conondrum. If you fail at investigating you are clearly also rationalizing everything you "learned" in your investigation. If you succeed, you realize it's an illusion... how? How are you not rationalizing? Are you not interacting with the illusion in some way?
I know it's a mechanic and fluff is secondary, but it relies on the disconnection between player and character to work, something that's not 100% present as assumed by the books in my game and won't work as written. It still ends up working more or less the same way: Either the player ignores the illusion and then there's no roll or the player interacts with the illusion somehow and it gets the description of what happens and, if an action was spent, the request for a roll.
External influences:
Well, it's hard to prove whether your interpretation or mine is correct. I believe, both are valid.
I think yours is an interpretation. Your has some part of the text to support your view (the target interacting with the phantasm.)
Mine is not really an interpretation of the text and more a ruling on the spell. I have not much in the way of text to point out other than the rationalization bit and that actually simply looking at something happening is enough to call "interacting", since you are recieving "updates". But again, it's totally fine to see outside sources as a way to have a foothold to investigate.
But in this case, I agree to Undrhil. For me, I take the wording of the spell description very verbatim. It says
"The target rationalizes any illogical outcomes from interacting with the phantasm." That means two things for me:
- The target rationalizes everything from interacting with the phantasm (and only with the phantasm).
- The target rationalizes inconsistencies only from interacting.
For this reason, I don't include
external influences as things being rationalized. It's not like you companion says
"hey, that's not real." and you instantly free yourself. But it might raise doubts and you still would have to investigate the effect. Keep in mind that other characters can't see the illusion, so they won't notice their companion doing weird stuff immediately. Depended from the illusion/situation that might take more or less rounds. But I think that's quite a fair deal.
Also, as I said before, this doesn't seem to be broken at all. There is no rule that the target has to take the illusion as
first priority. If it's caged - fine - then it still could cast spells from its position, use objects and attack with a ranged weapon. If you created an illusionary creature: Okay, but the target could also attack different creatures (as it would do normally).
In my opinion, the trigger of an investigation check is intended to be harder. Otherwise, WotC wouldn't have chosen such an
unusual and rare wording in the description (that most of the other illusion spells doesn't contain).
Absolutely not a broken spell anyway. And not really a question about if the person next to you won't be able to raise doubts about the actual existance of the illusion (people get scammed every day IRL, and that's pretty much convincing someone of something not real of being real, i see the opposite being very possible), it's the check itself nonsensical. You interact with the illusion and rationalize every illogical outcome. So, how do you realize that's an illusion? Mechanically, via check.
Fluff... you realize that's an illusion because something tells you "that's not real", going exactly opposite to what the spell says.
For me, it has to go between the "interaction" and the "rationalization" of the check at least. But i'm 100% sure you got it the first time i said it, so i'm just repeating myself and not bringing anything new on the table

It's totally fine to go for a different approach (a more RAW like, possibly, like yours seems to be)
Moving illusions:
In my view,
the object would only move, if it the "real version" of it would move as well:
- Manacles would move with the target, if they are not chained to the wall.
- A bridge would not move, because it wouldn't move in reality.
- For this reason, a gelatinous cube also would not move, because it wouldn't if it was real.
This is actually a very consistent and easy rule to prove, whether or not a created illusion would move with the target or not.
Just a small thing: Gelatinous cubes do move. Well, they CAN move, so probably would.
If a person moves beyond the reach of illusionary chains they break
Why would those break tho? The target can't change the illusion: It can't make a bear become a cat anymore than he can break the chains, both physically and mentally. Even the caster can't change the illusions (specific abilities being exceptions).
Also, the chains breaking is something that would make an illogical outcome logical, not a rationalization of an illogical outcome. That's not how the spell works.
I am a wizard. I summoned chains around your arms. What are you going to do about it?
DM: the caster will have to have experienced what it feels like to have been wrapped in chains - if you didn't spend time "practicing" with the effect, it counts as a poor quality spell.
I am a wizard. I summoned a box over your head. What are you going to do about it?
The victim may attempt to remove the box, or have a servant or have an npc remove it. Otherwise, it may delay the victim's action(s), or allow a saving throw to disbelieve.
There's no saving throw to dibelieve since any illogical outcome from interacting with the box gets rationalized right away. The only way to get rid of the spell once the first saving throw has been failed is to spend an action investigation the illusion and rolling a INT (investigation) check against spell DC. And the servant can't really do anything... the box is not there in the first place!