Count the days excluding Friday in every month using a date range.

Here you go...
Using your posted workbook, this regular formula returns the count of non-Friday dates for the month in Row_1 that are between the dates in Col_A and Col_B

Formula:

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Copy that formula across and down as far as you need.

See help on the NETWORKDAYS.INTL function for other options available.

Is that something you can work with?