You don't seem to account for null at all but it is hard to read
Code formatting ... the Community Version - Esri Community
<null> in python is None not a space
if a not in [None, " ", ""]:
    # then do something
Also, I see the python function, but don't see you calling it:
You need to call update() and pass the variables to it.
R_
Like the others have said, you actually have to call the function, and you should compare to None, not space.
x = [1, None, 2]
for y in x:
    print(y is None)
# False
# True
# False
Also, your function is needlessly complicated. Just use something like this:
# BFS_concat_trial = 
concat_if_not_empty([!Field1!, !Field2!, !Field3!, !Field4!, !Field5!, !Field6!], [None, "", " "], "/")
# code block
def concat_if_not_empty(values, empty, separator):
    """Concatenates non-empty values.
    values: list of values to concatenate
    empty: list of values considered empty
    separator: character used for concatenation
    """
    values = [str(v) for v in values if v not in empty]
    return separator.join(values)
