How to increase fanout of CMOS buffer
I would try to connect 10 buffers to the data source and then 10 74HC574 inputs to each buffer output. Think about a rack with a data bus to ten boards, each board has its own buffer for its internal data bus driving ten 74HC574. No problem with current and capacitance limits. Just avoid a common data bus to hundred HC574. It is possible to drive even a thousand 74HC574, connect 10 buffers to the data source and another 10 buffers to each buffer output of the first stage and then 10 74HC574 to each second stage buffer output.