First, we figure out the options for each clue, since they’re all pretty easy.

Down:
  1. GORGES or GRADES or IMAGES or ISRAEL
  2. BUS or HER
  3. FAT or ODE
  4. NITRIC or SHRIMP
  5. ERASE or LIGHT
  6. NANO or STAB
  7. GRATED or GUESTS or NEARED or NEEDED
  8. INVITED or INVOLVE
  9. NYAN or ONCE
  10. EDITED or EVADED or GRADED or GUITAR
  11. INVADE or INVEST
  12. GOLD or RICE
  13. EVIL or TEAM
  14. INVENTOR or INVOLVES
  15. CEASED or CREATE or NEEDLE or NUANCE
  16. TRUE
  17. ARABIC or HANDLE
  18. AUDIT or NEEDS
  19. INVENT or INVOKE
  20. INTERNET or LANGUAGE
  21. GEESE or GENUS or TONES or TREES
  22. ROCKS or SHELL
Across:
  1. FAMOUSSLOGAN or THEORETICIAN
  2. BEAD or HARD
  3. EATS or ODOR
  4. LABORATORIES or SOUTHEASTERN
  5. FALSE or TREAT
  6. EXTRANEOUS or TELEVISION
  7. CARGO or PIANO
  8. OVER or NEAR
  9. BEGAN or OCEAN
  10. GRID or ROAD
  11. IDLE or OPEN
  12. DUE or SET
  13. APPLE or ENDED
  14. DOUBLE or RECORD
  15. LATINO or MEXICO
  16. EAT or OFF
  17. AUGUST or HUMBLE
  18. APPEAR or NUMBER
  19. DOOR or EGGS
  20. INDEPENDENT or LIMITATIONS
  21. FACTO or THERE
  22. ERDOGAN or TIBETAN
  23. RAINED or SECOND
  24. EXCHANGE or STRAIGHT
  25. FINAL or TRIES

We could try to pick options and use them to fill the crossword, but it’s easy to get stuck with just one row/column left to fill and no words that fit. The flavor text tells us that yellow boxes are for extraction, so perhaps we should investigate the green and cyan boxes. If we examine which letters can go in the green boxes, we see that they all can be either F or T. We can also notice that the letters in the cyan boxes are forced, and there are 3 patterns:

A
N
O
I
N
D
R
V


It seems like this crossword is a circuit in disguise! When we draw out the circuit, it looks like this:


Since 28 Down is forced to be TRUE, we must find assignments for 1 Across, 7 Across, 34 Across, and 39 Across that satisfy the circuit. There’s only one assignment that works: false, true, false, false. So we use the words that start with F, T, F, and F respectively for those clues. From there, the crossword flows, with the result shown below:

1
F
A
M
O
U
S
S
L
O
2
G
A
N
O
3
H
A
R
D
E
G
4
F
O
U
R
E
A
5
S
O
U
T
H
E
A
S
T
E
R
6
N
7
T
R
8
E
A
T
I
R
T
A
R
S
I
9
E
X
T
R
A
N
E
O
U
10
S
11
C
A
R
12
G
O
T
U
A
13
I
14
O
V
E
R
15
B
E
16
G
A
N
N
N
S
U
V
C
T
17
I
18
G
R
I
D
19
I
D
L
E
20
S
E
21
T
N
O
T
T
E
22
I
V
L
A
E
A
N
23
E
N
D
E
D
24
R
E
C
O
R
D
25
M
E
X
I
26
C
O
V
N
E
27
E
A
28
T
29
A
U
G
U
S
T
30
A
P
P
E
A
R
N
R
R
D
S
T
U
A
31
I
U
E
O
E
B
N
L
32
D
O
O
R
I
V
33
I
N
D
E
P
E
N
D
E
N
T
34
F
A
C
T
O
N
K
T
35
E
R
D
O
36
G
A
N
E
E
R
37
S
E
C
O
N
D
N
H
U
E
E
38
S
T
R
A
I
G
H
T
L
39
F
I
N
A
L


We take the letters highlighted in yellow and anagram them as the flavor text indicates, giving SCRAMBLE.