First, we must establish a set of rules of which we operate with TOE.
So far you have not stated the "rules of TOE". Darwin did. As a matter of clarity, you ought to post them and reply to anything he stated that you oppose.
In general, most folks would agree that natural selection in particular and TOE in general represents the rule based systems of chemistry and physics as it pertains to life.
I don't think either chemistry or physics generally come to mind when we speak of Darwin's discovery. You may recall that he was a naturalist, and well grounded in that field. It's a long road from Mendel to Lavoisier or Newton. You haven't even begun to lay that foundation.
So, trying to evade abiogenesis and at the same time claiming TOE is based on a set of known rules is a contradiction.
Let's establish what the goalpost is, and who's running toward it, and who's running back. So far Darwin has scored all over the board. He has a huge global team, and they're scoring. Now: what does "evade" mean in the context of Darwin's theory? If you want to talk about abiogenesis, and how natural selection applies, you're in a bit of a quandry, since it lacks some the essential features. It's going to be unclear what having and excess of offspring means in abiogenesis, or competing for food. So, at a minimum you need to establish what it is you are claiming the theory says the is being upended. Then maybe we can figure out what you mean by "evasion".
Additionally, any "plant" creature must manufacture their own food. Otherwise, it is an animal.
What does this have to do with abiogenesis? You appear to be claiming that the first cell had a photosystem. That's sheer speculation, and probably one of the most problematic of choices at that.
Therefore, this LUCA must be capable of constructing carbohydrates which is food for the plant.
What does LUCA have to do with abiogenesis? What other energy sources are there besides the photosystem? Where are you on the timeline anyway? You seem to be jumping around.
You seem to be claiming that LUCA is the first cell. That's speculation, too, and also unlikely.
In order to construct carbohydrates, you must prove an implementation of an electron transport mechanism.
You need a lot more than that. But now you've jumped to some imaginary place on the timeline without establishing how you got there. What does this have to do with abiogenesis?
It's going to be hard to rise to some imagined standard of proof to compete with all of your speculation.
Currently, TOE has no such implementation.
Thankfully, TOE is not worried about electron transport. What does TOE have to do with this question? You haven't established that. You're jumping in time, and you're jumping fields, without laying the groundwork.
What are you claiming the TOE says, anyway?
No, you're just jumping to conclusions without establishing any rationale. We can get out the handbook and debug your arguments for common fallacies if you like. But I suspect you know that already.
All you have to do is to state the facts and refute them. But you're also going to have to concede to your errors. Otherwise we go around in circles.
Finally, I provided sufficient links from Oxford and others in which sulfur vents have been ruled out as a possible origin of life.
You're exaggerating your own source. It never said anything about the origin of life. It was speaking only about a hypothetical LUCA. It does not reach back earlier than the presumed LUCA, and it says nothing about the origin of life whatsoever. This is where you're jumping around on the timeline again. Remember, this was an extremely long geologic era. A lot of stuff happened -- a lot more than what this one study was addressing.
You seem to believe that the photosystem and sulfur exploit were the only energy sources at some era still undefined. You are leaving out all the other exploits, and you haven't discussed the state of evolutionary biology concerning the rise of organelles, much less the photosystem.
That's what I meant about jumping to conclusions. Oversimplification is an easy way to short circuit logic. It's the hallmark of fallacy. Not a good strategy for arguing science.