An LHG would let you receive very weak signals and also transmit at a very high gain. The answer to your question is that in reality it depends on a lot. Most importantly it depends on two devices being able to speak and hear at a good level. The router in a coffee shop speaks at a low level and also hears at a low level too, this is opposite of a LHG which speaks at high levels and can hear a very very wide range of levels. I don’t think you could successfully establish a reliable bidirectional connection, but you could certainly have a lot of luck with simply receiving weak signals from the normal router,
The reason the coffee shop router may not be able to receive a signal from an LHG is because the antennas on the router are not as big as the LHG and depending on your distance and obstacles the signal may fade away before it reaches the router.