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Okay, here's how we really did it.

The cryptic in part 3 can be solved normally. The gimmicks are that words extend outside the grid (without letting you know by their enumeration) and that each clue has a word that must be "beheaded" before entry. Reading the letters extending outside gives you OVERSIZE.

  The rebus words in part 6 can be identified. We don't have the pluses or minuses, but there's a unique way to combine everything so the subtracted words are substrings of the added words. The remaining letters spell EGOTISTICAL.

The "heads" of the beheaded words from part 3 spell "opposite of pro". Part 7 seems to have trigrams (three-letter sequences) taken from other parts of the puzzle, and there's a piece mentioned in part 2. This means the word must be of the form [???][CON][???][???] or [???][???][CON][???], with the bracketed parts being wordlike. The only thematic answer here is DISCONTENTED.

At this point a solver would realize that the seven parts are themed around the 7 Deadly Sins, and the answers describe someone who overindulges in that puzzle's sin. The meta explanation mentions two different orderings, and it seems that the answer lengths increase by 1 as you go down from part 1 to part 7. With these heavy constraints and a bit of additional info from the descriptions, we can now figure out the other answers.

Part 1 has "metal in it"; the only word meaning "overindulging in greed" with a metal inside is STINGY.
Part 2 can't be figured out on its own: it can only be fully backsolved from the meta answer.
Part 4 has a supposed typo, and the only synonym of "angry" that has an A after an R (where the A could plausibly be typoed to an I) is IRRITABLE.
Part 5 mentions "inserting into a woman" and is pretty clearly the lust puzzle, so the answer LASCIVIOUS fits (having "LASS" around the outside).

The first letters of the answers spell S?OILED, so the answer is SPOILED.

Okay, here's how we really did it.

It's initially unclear how to attack the puzzle as a whole, but some of the parts are more direct than others, so we can start with those...

The cryptic in Part 3 can be solved mostly normally, however, there are two gimmicks involved.

Firstly, each clue has a word that must be "beheaded" before the clue will work. Secondly, some clues provide the incorrect enumeration (it becomes clear why when coming to fit them to a grid):

Grab a circle holding tin counting machines (5): AB A CI
Court-worth Nascar Jr. heating 60s diet cola (6 7): DA(TAB)LE
Every one estrange ache (3 4): EACH*
Before person orating epitaph, initially (3): (PRE)_
Price label preceded pistol (3): GAT<
Listen to successor of musical with "Paint Got No" (3 4): HAIR ("heir")
Bad ozone rig resetting, like a scale. (6 7): ZEROING*
Adept shooters hold profundity (5): DEPT H
Saying "Great scott!" irises after best grade (5): A+DAGE<
Captain Morgan rival king, for example, tenters one who likes both (6 7): B(A+CARD)I
EA hot new vow (3 4): OATH* (used in one of OP's previous puzzles)
Top of open hat (3): CAP (ddef)
Brilliant TV brain faltered (6 7): VIBRANT*
Pleaders of law enforcement indight gravely heinous actress Vivien (5): (LEIGH)_
Low digit ripped right rout (3): TO(-r)E
Chips front off middle of Omen (3 4): (-c)HIPS

Treating the clues as coming in standard across/down order, it's relatively easy to fit them to the grid, as shown above, but with extra letters (where longer than the given enumeration) extending outside the grid. Reading these off clockwise from the top, gives the answer: OVERSIZE.
Also note that the extra letters in the clues, when taken in order, spell opposite of pro, giving us CON (to be used later).

The cryptic clue/rebus words in Part 6 can be identified from the errata/advice. Some were relatively obvious/limited, and once the pattern emerged the rest became simpler. Specifically:

&lit from Wikipedia => EGG
"Emotion" cluephrase => EMOTIVE
Sparkly and pretty => GEM
Arthropod developmental phases => INSTAR
Technically an artery => VEIN
Final/physical/oral exams => MEDICAL
Relating to "night vision" => ARMED

Using the above we can see that the words fall into an overlapping pattern:
EG(G EM)OTI(VE IN)ST(AR MED)ICAL
So, whilst we don't have access to the "pluses and minuses on the flag" mentioned in the last step of this part, it doesn't take too much to realise that we can subtract the overlapping parts (EGG + EMOTIVE + INSTAR + MEDICAL - GEM - VEIN - ARMED) such that the remaining letters spell EGOTISTICAL.

Thanks to our earlier work, we have a partial lead on Part 7:

The clues here talk about trigrams (three-letter sequences) and "stolen" parts, which we can assume are stolen from other parts of the puzzle. We happen to have found one of these in part 3 (the CON we derived from the extra clue letters). There's also mention of another one in Part 2 ("one part is used for a later puzzle"). This means the word must be of the form [???][CON][???][???] or [???][???][CON][???], with the bracketed parts being wordlike. The only thematic answer here is DISCONTENTED.

At this point a solver would realize that the seven parts are themed around:

The 7 Deadly Sins, and the answers describe someone who overindulges in that puzzle's sin. The meta explanation mentions two different orderings, and it seems that the answer lengths increase by 1 as you go down from part 1 to part 7. With these heavy constraints and a bit of additional info from the descriptions, we can now figure out the other answers.

Part 1 has "metal in it"; the only word meaning "overindulging in greed" with a metal inside is STINGY.
Part 2 can't be figured out on its own but PASSIVE is all that will work when backsolving from the meta answer. However, it is still thematically described in the clue ("allows us to be extra lazy").
Part 4 has a supposed typo, and the only synonym of "angry" that has an A after an R (where the A could plausibly be typoed to an I) is IRRITABLE.
Part 5 mentions "inserting into a woman" and is pretty clearly the lust puzzle, so the answer LASCIVIOUS fits (having "LASS" around the outside).

The first letters of the answers spell S?OILED, so the answer is SPOILED.

Screenshot of the wrapup from your latest puzzle:

(Huh, some of the less useful text turned red when I screenshotted it! That's strange.)

Screenshot of the wrapup from your latest puzzle:

(Huh, some of the less useful text turned red when I screenshotted it! That's strange.)

Okay, here's how we really did it.

The cryptic in part 3 can be solved normally. The gimmicks are that words extend outside the grid (without letting you know by their enumeration) and that each clue has a word that must be "beheaded" before entry. Reading the letters extending outside gives you OVERSIZE.

The rebus words in part 6 can be identified. We don't have the pluses or minuses, but there's a unique way to combine everything so the subtracted words are substrings of the added words. The remaining letters spell EGOTISTICAL.

The "heads" of the beheaded words from part 3 spell "opposite of pro". Part 7 seems to have trigrams (three-letter sequences) taken from other parts of the puzzle, and there's a piece mentioned in part 2. This means the word must be of the form [???][CON][???][???] or [???][???][CON][???], with the bracketed parts being wordlike. The only thematic answer here is DISCONTENTED.

At this point a solver would realize that the seven parts are themed around the 7 Deadly Sins, and the answers describe someone who overindulges in that puzzle's sin. The meta explanation mentions two different orderings, and it seems that the answer lengths increase by 1 as you go down from part 1 to part 7. With these heavy constraints and a bit of additional info from the descriptions, we can now figure out the other answers.

Part 1 has "metal in it"; the only word meaning "overindulging in greed" with a metal inside is STINGY.
Part 2 can't be figured out on its own: it can only be fully backsolved from the meta answer.
Part 4 has a supposed typo, and the only synonym of "angry" that has an A after an R (where the A could plausibly be typoed to an I) is IRRITABLE.
Part 5 mentions "inserting into a woman" and is pretty clearly the lust puzzle, so the answer LASCIVIOUS fits (having "LASS" around the outside).

The first letters of the answers spell S?OILED, so the answer is SPOILED.

Screenshot of the wrapup from your latest puzzle:

(Huh, some of the less useful text turned red when I screenshotted it! That's strange.)

Screenshot of the wrapup from your latest puzzle:

(Huh, some of the less useful text turned red when I screenshotted it! That's strange.)