"Simply Not" vs. "Not Simply"

Yes. Collins brings out these differences:

simply [adverb] [adverb before verb {phrase}, adverb adjective]

You use simply to emphasize what you are saying. [emphasis]

• This sort of increase simply cannot be justified.
• So many of these questions simply don't have answers.
• In a poll of those leaving the theatre and nine out of ten thought it was simply marvellous.

This 'simply' is a pragmatic marker, relating to the whole statement:

'Let me put it simply: one does not walk into Mordor.' or

'One does not walk into Mordor. It's as simple as that.'

......

The second positioning rather uses the senses (and I'd say fused):

simply [adverb]

(1) in a simple [uncomplicated, undemanding] way

(2) merely, only [just]

This 'simply' addresses the walking into Mordor:

'You can't just walk into Mordor as if you're out for an afternoon stroll.'

This is more the traditional adverb role, and note that the adverb is closer to the verb 'walk'.

I agree, the first statement offers less hope.

1. One simply does not walk into Mordor.

This means, "Simply said; One does not walk into Mordor."

It is prescriptive. In your example, it seems to prohibit walking into Mordor at all - maybe you should fly instead, or go on a horse.

Traditionally this form describes etiquette. For example, "One simply does not wear brown shoes with a blue suit."

A conditional example might be:

"One simply does not walk into Mordor without making obeisance to Sauron. It would be bad manners!"

or

"One simply does not walk into Mordor wearing sandals and shorts. It's not the done thing"

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2. One does not simply walk into Mordor.

This is the form I would expect from the context. It implies that a simple walk will not be sufficient to gain access. Maybe you need stealth or an army or an invisibility cloak.

One simply does not walk into Mordor.

"simply" is modifying "does not". The inability to walk into Mordor is simple. That is, walking into Mordor is not possible at all.

One does not simply walk into Mordor.

"simply" is modifying "walk". If one is to enter Mordor, one does not do so by simply walking. This leaves open the possibility that one might be able enter Mordor, if one does something more complicated than just walking. For instance, one might be able to enter Mordor, if one walks into Mordor, and they do so while being guided by a being hundreds of years old, and a gigantic army distracts Sauron, and an encounter between Sauron and your allies in a Palantir convinces him that the One Ring is elsewhere.

"One simply does not walk into Mordor" -> One does not walk into Mordor. It's preposterous to do so.

"One does not simply walk into Mordor" -> One must do something more complicated than walking if one wants to go to Mordor.

Bonus!

"One does not walk simply into Mordor" -> This is kind of a bad example because using "simply" in this grammar form is a bit of a stretch; a better example might be "One does not walk slowly into Mordor". The meaning being one can enter Mordor, but walking slowly is not a good way to do so; you're likely to get ambushed by orcs. Walking quickly, however, may be acceptable.

1. One simply does not walk into Mordor.

Rephrase: Not possible to walk into Mordor, pure and simple.

Meaning: The information of the impossibly of walking into Mordor is simple.

2. One does not simply walk into Mordor.

Rephrase: You can't just walk in (you need to get a pass first).

Meaning: It's not easy to get into Mordor.