Just to clarify, they are using a rubberized mallet when hitting the boards (though they do seem to be making slight 'delamination' type marks along the edges here and there - at least, I assume that's what the marks are from. (?)

The block is intended to reduce damage to the edges that will show after installation.

It is even more important with pre-finished material since there is no sanding/finishing step to possibly remove any slight damage.

Thanks - so I'm not sure if I should tell the installers I didn't appreciate them not using one? Is it standard practice for a "pro" not to use them?

If the manufacturer installation guideline says to use a tapping block, then they are installing your floor improperly. What does the manufacturer say?

It is fine to shave a little off the clic profile and then to glue the pieces. That is how it is done with most floating panels to secure them when encountering work arounds.