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Sounds like the points are grounded. It is possible to easily and accidentally ground the points by improperly installing the lead wire to the points. Check carefully to be sure any insulators are in the correct position in relation to the lead wire.
If power is flowing through the points and no spark is being generated at the plug ASSUMING the plug is properly grounded on the engine, the coil or its connections are at fault. FWIW, you don't really need the plug for the test, just arrange to hold the spark plug end of the coil wire within about 1/8 inch from the engine block.
When the points are OPEN there can be no current flow through the coil because the open moveable point has no connection to ground.
If the points do not close due to incorrect adjustment there will be no ignition. Check this with the multimeter to be sure they are ELECTRICALLY closing, not that they APPEAR to be closing.
Glad it's fixed.
Used to have an Ariens snow blower...
Since I replaced everything except the points & plug lead, I don't know WHAT the original problem was.
Carburetor Paul B