Used to use 10-40 in my Briggs 2HP's. Nothing bad happened.
Not recommended by B&S
Tecumseh manuals specifically state do not use 10w40. Reasons are obvious.
ENGINE OIL
TECUMSEH FOUR CYCLE ENGINES REQUIRE THE USE OF A CLEAN, HIGH QUALITY DETERGENT
OIL. Be sure original container is marked: A.P.I. service "SF" thru "SJ" or "CD".
TECUMSEH RECOMMENDS USING ONE OF THE FOLLOWING FOUR CYCLE OILS THAT ARE SPECIALLY
FORMULATED TO TECUMSEH SPECIFICATIONS.
DO NOT USE SAE10W40 OIL.
This is not to say that some manufacturers do recommend it.
If they do, then by all means, use it.