OP: "Can a monk choose to eat only an animal or plant that had died
Yes. (The monk could make that choice, but as was pointed out by others the monk's should not be picky or fussy or instruct/command lay folks in what they should give them.)
"This either by scavenging on his own or by waiting for others to
offer this to him?"
Yes. (The monk could do this, but then the monk would be picky/fussy which could run them afoul of vinaya rules.)
OP: "Yet a monk can avoid all this simply by waiting for someone to put the
dead animal in his bowl."
OP: "The morality of this doesn't make sense."
The monk would be faultless as it is simply not true that the monk acted as a condition for the killing of the animal. In your hypothetical, the animal was killed and dead before the monk had any contact with it. The animal was killed and dead before the monk had any contact with the person who killed it. Saying the monk is at fault is akin to misunderstanding how the law of cause and effect works. To say that the monk was the cause or a necessary condition for the killing effect would be akin to saying the effect preceded the cause. This is a logical error and not how the world works.
The law of karma is not an impersonal arbiter of universal justice meting down punishment on sinners who have transgressed some moral rules. To suppose otherwise is to believe that karma is truly existent. It is not.
Cause and effect is described over and over in the Pali suttas as:
iti imasmiṁ sati idaṁ hoti; imassuppādā idaṁ uppajjati; imasmiṁ
asati idaṁ na hoti
Which has variously been translated as:
"When this exists, that is; due to the arising of this, that arises.
When this doesn’t exist, that is not; due to the cessation of this,
"When this exists, that comes to be; with the arising of this, that
arises. When this does not exist, that does not come to be; with the
cessation of this, that ceases."
"When this is, that is. From the arising of this comes the arising of
that. When this isn't, that isn't. From the cessation of this comes
the cessation of that."
"If this is, that comes to be; from the arising of this, that arises;
if this is not, that does not come to be; from the stopping of this,
that is stopped."
The Buddha said this again, and again, and again, and again, as he taught the Dhamma to people. Your contention boils down to if the monk stopped accepting dead meat placed in his bowl, then all the suffering that resulted in the killing of the animal would cease. However, we know this is not the case simply by virtue of the fact that the killing of the animal and all that suffering that necessarily followed preceded the monk accepting the dead meat in his bowl.
To see this clearly, try putting your contention into the words of the Buddha:
"With dead meat in monk's bowls, killing of animals comes to be; from the
arising of dead meat in monk's bowls, killing of animals arises; When
dead meat is not in monk's bowls, killing of animals does not come to be;
with the cessation of dead meat in monk's bowls, killing of animals
That is just clearly not true. Were the dead meat not in the monk's bowl, the animal would still be every bit as dead from the killing and the suffering that necessarily entails would follow.