count # of unpaid months in data, replacing pivot that counted the unpaid months

Seems to me that what you are attempting to do is necessarily complicated.
To demonstrate a part of what I believe would be needed a range (K:V) has been populated showing for which months the corresponding sources and lists have values using:

Formula:

=IF(COUNTIFS($A$2:$A$129,K$1,$B$2:$B$129,$B2,$C$2:$C$129,$C2,$D$2:$D$129,$D2,$E$2:$E$129,$E2,$G$2:$G$129,$G2,$H$2:$H$129,$H2,$I$2:$I$129,$I2),1,"")

The count is in column W
I feel that you would then need to make a table to get the sums per month showing ones in the first added table and then get the average.
This is not to mention that you might need a further table that would display the unique sources/lists along with counts and averages so that you aren't having to look through multiple rows of duplicate information.
I would think a better choice would be to display the pivot table in tabular form, along with the count and average columns as modeled.
Note: It appeared that originally the columns for average were supposed to only average values for certain months. The column for average that is being modeled in column T has an adjustable range based on the values in T1:T2

Formula:

=IFERROR(AVERAGEIFS(F6:Q6,F$5:Q$5,">="&T$1,F$5:Q$5,"<="&T$2),0)

Let us know if you have any questions.