There are certainly amps (mostly receivers) rated for only 8 ohm loads (or higher, obviously).
Do I think an amp cares about the load it sees? Most certainly. If not, there would be no need for stouter power supplies, multiple output transistors, more heat-sinking, voltage regulation, current limiting circuits, etc.
Not sure why you are dwelling on speakers rated at 6 ohms.
What you do is of course your choice, but for those less interested in the details the proper approach is to use an amp rated for an impedance equal to or lower than the speaker's rated impedance.
Even if no damage ensues from doing otherwise, the less able the amp to deliver current demanded by the load, the less linear its output.