Testing my 2-wire home receptacles with a Southwire model 50020S-A
Use the Test Button on the outlet!

"GFCI testers" do indeed rely on a ground to provide a "controlled leak" past the GFCI so that it trips. The test you really need to be doing to test the GFCI protection is pushing the test button on the GFCI itself, as that leaks a bit of current line-hot to load-neutral (or vice versa) and thus trips the GFCI.  Also, the indication you get from the testing device when the button is pushed is indeed no good due to that open ground.