There can be totally appropriate and legitimate overlap with Perception and Investigation. It's a case-by-case, dependent on the situation basis...
The problem with taking this approach to extremis (and not going into your further argument, just this part) is that it becomes impossible to build a character who is good at certain things you want them to be good at. Say I am making a dwarf who is a skilled stonemason, and I want them to be good at noticing unusual stonework, including traps and secret doors. If you use a simple rule like "Perception notices creatures, Investigation notices things" it is very clear; my character should invest in Investigation.
This is also why I dislike systems where the GM arbitrarily assigns an attribute+skill combination for each task; its impossible to design a character and know they are actually good at something. A sensible approach to my mind is to say that you notice secret doors using one specific skill, say Investigation, and investigation is based on Intelligence. So the roll is Intelligence + Investigation. But, in a specific situation, the GM may give the player the
option to use other combinations of skill+attribute. Say this is a secret door the PC knows is on a certain wall, but the wall is a mile long - allow the player the
option to roll Con+Investigation to represent the energy to look over every little bit of wall and not lose focus. But don't take away the standard option.
This leads me to the claim I made in 4E as a joke. Let me play a dwarf, and I can make very roll in every skill challenge ever using Stamina (or whatever the skill was called), as long as the task is measured in time longer than rounds. [I mention this as a joke more than as a challenge.

]
I need to convince the king? I hold a loooong oration, quoting every generation of my clan since time immemorial until the king simply gives in due to fatigue. To interrupt a dwarf is inexcusable!
I need to win over the kings diplomats at dinner? I drink them under the table.
I need to disarm a trap? I very carefully and above all methodically line out every possible option for how the trap could work, never losing my concentration.
I need to travel the wilderness? I just never give up.
And so on, ad nauseum.