ALSO A PARTIAL "SOLUTION"
STILL A PARTIAL SOLUTION
I went another route since using traditional puzzling techniques didn't get me very far or else proved to be too much work. I took the title at its face value and got the following:
The way I got this solution?
Sometimes force is the answer
Welp, maybe FORCE really is the answer! Let's check it against the ridiculously high-tech formula $-05↓ -10↑ -14↑ +09↑ +03↑$ and see what we can find...
'F'can be represented as (6), so we subtract 5, and get 1; the arrow indicates lower case, so the first character is 'a'...$$F \rightarrow a\\O \rightarrow E\\R \rightarrow D\\C \rightarrow L\\E \rightarrow H$$
So the above image lives at Stack imgur ID
aEDLH. Unfortunately, you don't get anything from doing the same thing with the letters on the card. SVTDP $\rightarrow$ nLFMS = nada.
Okay, fine. Let's take this in another direction. Maybe...
Let's go ahead and BRUTE FORCE this sucker. We'll be working with the sticky note in this scenario. Let me tell you, it's quite difficult to find an anagramming system that will allow you an input with so many characters, and for good reason! There are uncountable possibilities with so many letters to work with. However! You can get a few rather relevant words out of it if you really look.
A set of words with full usage of all letters in
TEHEMEHTREATNCIAGNELEERTC is: "
GENERATE LETTER MACHINE TECH". You can also find potentially useful partial anagrams such as
INTERCHANGE... I'm going to post more findings at the bottom of the post.
Disregarding the bigger picture for now and getting down to the minutiae...
The individual letter sets can be made into real words by feeding them through the machine twice on what I believe is Mode 1. Taking TEHEM and rearranging its letters based on the pipe tracks ($12345 \rightarrow 15423$) gets you TMEEH. Feeding the result through again gets you THEME. $$TEHEM \rightarrow THEME\\EHTRE \rightarrow ETHER\\ATNCI \rightarrow ANTIC\\AGNEL \rightarrow ANGLE\\EERTC \rightarrow ERECT$$
Okay, that's all well and good. Let's try this on our keycard before doing the math. SVTDP $\rightarrow$ SPDVT $\rightarrow$ STVPD, apply the algorithm to get
nJHYG, aaaand ... no such image. Dangit.
Fun with anagrams
ELECTRIC NETHER GATE METHANE is a full solution. We've got powerful farts, people.
This alien world also features a rather exclusive golf club or restaurant:
GENTLEMAN ARCHITECT TEE HERE/
ARCHITECT GENTLEMEN EAT HERE
I think they've also got Washington's mythical dentures on display:
ARCHAIC GENTLEMEN TREE TEETH
They've got a strange take on the "three guys walk into a bar" joke...
TIGER CHEETAH CAMEL ENTER NET
Apparently they don't have a word for "octopus"...
THE ERR EIGHT TENTACLE MENACE
Maybe Sue Sylvester got a boob job?
GLEE TEACHER TEAT ENRICHMENT
Thanks to a large bounty offered/awarded by OP (thanks, Avigrail!) this puzzle made its way into my notifications and back onto my radar, complete with some fresh updates (directions on how to use the dang thing) and subsequent analysis by blurry - thank you, internet.
So, coming off of all the work done by us as a whole, we've established that we should...
- Set to Mode #2 (letter order) and run it twice.
TEHEM EHTRE ATNCI AGNEL EERTC $\rightarrow$
THEME ETHER ANTIC ANGLE ERECT
- Set to Mode #1 (word order) and run it twice.
THEME ETHER ANTIC ANGLE ERECT $\rightarrow$
THEME ANTIC ETHER ERECT ANGLE
- Set to Mode #3 (character [?] shift) and run it.
THEME ANTIC ETHER ERECT ANGLE $\rightarrow$
THE MEANT ICE THERE RECTANGLE???
Well, shoot. I still don't know what to do and my brain is fried. I'm sure it has something to do with feeding all the words through at once and having it come straight out as noise, ...so you have to reinterpret it as homophones? "The man took either a rectangle"? Anyway, "rectangle" looks like a pretty valid option, whatever you're supposed to do.
This puzzle keeps teasing me from the sidelines. I found a Thing today. I don't know if it is the Right Thing. But gee golly, anything that isn't that blasted "does not exist" message is a welcome change.
Last time on "Avigrail is Too Smart for All of Us": ($12345 \rightarrow 15423$) and Mode #3 does *shrug emoji*
From our list of words coming off of Mode #2, we have
THEME ANTIC ETHER ERECT ANGLE so I tried that same number sequence to pull out individual letters - 1 from the first, 5 from the second...
TCERG. Now we complete the last two steps and run that sequence through Mode #2 twice again for TCERG $\rightarrow$ TEGRC. Run it through the algorithm to get...
Now... imgur does INDEED have a result for this string.
Assuming that this is REAL, the 5-character string is what we want (it has been so far, right?) let's do some more messing around...
MMIJB $\rightarrow$ hCUSE
... aaaand, nothing. Fudge. Nothing for running the 5 through Mode #2 first either.
MBJMI $\rightarrow$ hRVVL
MIMBJ $\rightarrow$ hYYKM