Rob, I'll refer to the Tecumseh engine and transmission manual (a 1,400 page document available in the Manuals section of this site).

The manual describes the type of breather you have as a "side mounted breather", and indicates it installs with the large port and the reed toward the top, so the tiny drain holes are in the bottom, for obvious reasons.

That may well be the answer to your problem, as you suggested. Just check that the tiny drain holes are in the bottom horizonal surface of the breather, not the top.

Here are a couple of excerpts from the manual about trouble-shooting for oil passing through the breather - note the emphasis on installing the valve with the drain holes facing down:
The Breather is a sealed assembly. If it is defective, replace the entire assembly. The rubber tube on the outside is not part of the assembly. Install the breather with the drain holes toward the base of the engine. Always use a new gasket. The retaining screws must be tight.
If oil is being pumped out the breather:
1. Check for excess speed.
2. Operating at excessive angles.
3. Loose oil filler cap.
4. Worn rings.
5. Excess blow-by.
6. Breather body installed up-side down.
7. Overfilled with oil.

All of that is standard stuff, except the warning about the loose filler cap.

Here is another troubleshooting guide for breather oil from later in the manual:
BREATHER PASSING OIL
1. Check the oil level, make sure the engine is not overfilled. Also verify that the viscosity rating on the container of the oil being used is to specification.
2. Check the angle of operation. Avoid prolonged use at a severe angle.
3. Check the engine R.P.M. setting for excessive R.P.M. using a vibratach or other tachometer and compare it to the R.P.M. settings found on microfiche card # 30 according to the engine model and specification number. Adjust the high and low R.P.M. as necessary.
4. Check for leaking or damaged gaskets, seals, or "O"-rings. External leaks may not be evident; however, the leak may prevent the engine from achieving a partial crankcase vacuum.
5. Check the breather for damage, dirty condition, or improper installation. The oil return hole(s) must face down.
6. Check the engine compression using a compression tester. If the engine has weak compression, determine the cause of weak compression: worn rings, leaking head gasket, or leaking valves. Follow the compression tester's procedure.