One thing that works better than the valve core wrenches to remove the valve core are the dust caps that are metal and have the end forked like the one end of the tool you bought. I've found that these caps with the tool on them go into all valve stems very easily and work great; whereas, the valve core tool often is very tight or almost impossible to get into the stem. The tool will remove really, really tight valve cores better, IF you can get it in there and on them.
I removed stems on 4 tires today with a valve cap. My dolly tires had gone flat. To get them to reseat on the rims requires removal of the valve stem to get lots of air in to seat the tire bead onto the rims (aided by some fluid squirted onto the beads). Then reinsert the valve core and inflate.
Then I fixed one flat, slimed it, and then slimed the other front tire. Had to remove the cores for that, too.
The big (32 oz) slime container had a valve core cap with the tool on the end underneath its plastic cap. I had one anyway.
Ralph