. Can ARM-USB-TINY-H put the SRST pin into a high impedance state?
. What is the resistance value of the pull-resister to the SRST pin in ARM-USB-TINY-H?
In ARM-USB-TINY-H SRST has big pull-down 1M. It is good to have some pull-up SRST at your target (10k would be good value for the pull-up at the target, but no more than 50k).
SRST can also toggled via software means since there is transistor routing it to the FT2232HL, try with the software options provided in OpenOCD.