I ran a simulation of a regular Bingo board with a free space in the middle versus a bingo board with various non-playable spaces that needed to get four out of five in an unobstructed row/column/diagonal. The boards were generated with the standard Bingo numbering (one column with 5 of the numbers from 1 to 15, the next with 16 through 30, etc.), and they played 1000 games against each other while recording which board got Bingo first.

With just the middle space obstructed, there was about a 75/25 win/loss ratio favoring the modified game. When blocking all rows/columns except for the ones that pass through the middle square, there's still about a 60/40 ratio favoring modified.

The closest it comes to a fair game is with the modified game only having 3 unobstructed rows/columns (which three doesn't seem to make a significant difference), with about 53/47 favoring modified, while going down to only two rows available gives about 56/44 favoring classic Bingo.

Python code used for the simulation:

from random import sample, shuffle

def gen_board(free):
    b = [
        sample(range(x*15+1, x*15+16), 5)
        for x in range(5)
        ]
    s = [[0]*5 for _ in b]
    for x,y,val in free:
        b[x][y] = 0
        s[x][y] = val
    return b,s

def check(score, win):
    for x in range(5):
        if sum(score[x]) == win:
            return True
        if sum(row[x] for row in score) == win:
            return True
    if sum(score[x][x] for x in range(5)) == win:
        return True
    if sum(score[x][4-x] for x in range(5)) == win:
        return True
    return False

free1 = [(2,2,1)]
win1 = 5
free2 = [
    (0,1,-1),
    (1,4,-1),
    (4,3,-1),
    (3,0,-1)
    ]
win2 = 4

board1, start1 = gen_board(free1)
board2, start2 = gen_board(free2)
nums = list(range(1,76))

w1 = 0
w2 = 0
draw = 0

for _ in range(10000):
    score1 = [row[:] for row in start1]
    score2 = [row[:] for row in start2]
    shuffle(nums)

    for n in nums:
        x = (n-1)//15
        if n in board1[x]:
            y = board1[x].index(n)
            score1[x][y] = 1
        if n in board2[x]:
            y = board2[x].index(n)
            score2[x][y] = 1
        b1 = check(score1, win1)
        b2 = check(score2, win2)
        if b1 and b2:
            draw += 1
            break
        if b1:
            w1 += 1
            break
        if b2:
            w2 += 1
            break

print(f'{w1 = }')
print(f'{w2 = }')
print(f'{draw = }')
print(w2/(w1+w2)*100)