A less technical solution:

We can (try to) make a symmetric solution that makes all numbers equal (where the 6 variables signify how often a cell is chosen):
abcba
bdedb
cefec
bdedb
abcba
For the total we get: T = a+2b = a+b+c+d = 2b+c+e = 2b+d+2e = 2d+c+e+f = 4e+f From which we can extract equalities:
 (1+3) a = c+e
(1+2) b = c+d
(3+4) c = d+e
(5+6) 3e = c+2d
(4+6) f = 2b+d-2e = 5d

Any positive integer solution of this must be a multiple of f=10, d=2 etc, leading to a total that is a multiple of 22.

Any asymmetric solution can be made symmetrical by adding up all 8 (horizontal vertical and diagonal) reflections so 8 times any solution must lead to a multiple of 22. Thus any single solution leads to a multiple of 11. 2020 is not a multiple of 11.

A less technical solution:

We can (try to) make a symmetric solution that makes all numbers equal (where the 6 variables signify how often a cell is chosen):

abcba  
bdedb  
cefec  
bdedb  
abcba  

For the total we get: T = a+2b = a+b+c+d = 2b+c+e = 2b+d+2e = 2d+c+e+f = 4e+f

From which we can extract equalities: 

(1+3) a = c+e
(1+2) b = c+d
(3+4) c = d+e
(5+6) 3e = c+2d
(4+6) f = 2b+d-2e = 5d

Any positive integer solution of this must be a multiple of f=10, d=2 etc, leading to a total that is a multiple of 22.

Any asymmetric solution can be made symmetrical by adding up all 8 (horizontal vertical and diagonal) reflections so 8 times any solution must lead to a multiple of 22. Thus any single solution leads to a multiple of 11. 2020 is not a multiple of 11.

A less technical solution:

We can (try to) make a symmetric solution that makes all numbers equal (where the 6 variables signify how often a cell is chosen):
abcba
bdedb
cefec
bdedb
abcba
For the total we get: T = a+2b = a+b+c+d = 2b+c+e = 2b+d+2e = 2d+c+e+f = 4e+f From which we can extract equalities:
(1+3) a = c+e
(1+2) b = c+d
(3+4) c = d+e
(5+6) 3e = c+2d
(4+6) f = 2b+d-2e = 5d

Any positive integer solution of this must be a multiple of f=10, d=2 etc, leading to a total that is a multiple of 22

!Any asymmetric solution can be made symmetrical by adding up all 8 (horizontal vertical and diagonal) reflections so 8 times any solution must lead to a multiple of 22 Thus any single solution leads to a multiple of 11. 2020 is not a multiple of 11

A less technical solution:

We can (try to) make a symmetric solution that makes all numbers equal (where the 6 variables signify how often a cell is chosen):
abcba
bdedb
cefec
bdedb
abcba
For the total we get: T = a+2b = a+b+c+d = 2b+c+e = 2b+d+2e = 2d+c+e+f = 4e+f From which we can extract equalities:
(1+3) a = c+e
(1+2) b = c+d
(3+4) c = d+e
(5+6) 3e = c+2d
(4+6) f = 2b+d-2e = 5d

Any positive integer solution of this must be a multiple of f=10, d=2 etc, leading to a total that is a multiple of 22.

Any asymmetric solution can be made symmetrical by adding up all 8 (horizontal vertical and diagonal) reflections so 8 times any solution must lead to a multiple of 22. Thus any single solution leads to a multiple of 11. 2020 is not a multiple of 11.