Looking for multi-channel event counter IC to verify switch operation
Your best bet is to use a cheap FPGA development board.  Here is a list of them at Digikey, sorted by price.  There are several in the $30 range.

Get one with enough I/Os to handle the number of signals that you need to count.  For example, the Lattice MACHXO2 board has over 100 I/Os that go to 0.1" pin headers.  It is programmable over USB and the development software is free to download.

As for programming the FPGA, instead of trying to have a large number of external clock inputs directly drive a bunch of counters, it would be best to use a fixed clock, sample the external inputs, and increment the counter when a rising edge is detected.  A rising edge is detected when a 0 sample is followed by a 1 sample.  The clock frequency must be at least ~4X than the input frequency and the sampled inputs should go through a couple levels of registers to avoid any metastability issues.

You can also use the FPGA logic to implement an SPI or I2C interface to read counters and status, reset the counters, etc.