Not mechanically minded but can usually find solution to my problem if I can see or work out what is wrong; however today while out mowing my ride on started making a noise like a cog or gear was slipping.
Turned it off, had a look and a small hose above the throttle cable had come loose and oil was spraying out.
Left it for an hour or so after i had re-attached it, checked oil, started again, the clunking sound was there initially, but then stopped after a few seconds, but the engine did not idle at it's normal speed, it was much, much higher and more exhaust smoke than usual and even reducing throttle power doesnt reduce the engine revs.
You should not be able to pull up on the throttle shaft more than a few thousands of an inch. Probably what has happened is the screws have come out of the throttle shaft that hold the throttle butterfly. With a loose butterfly, you have no throttle or governor control. Do not run the engine until repaired!
The screws probably got sucked into the the combustion chamber and if you're lucky, wound up in the muffler. You can verify missing screws or loose butterfly by removing the air filter and look inside the carb throat to the rear. I have even seen a throttle shaft wear completely through and break on a couple old Kohlers.
Let us know what you find. You very well might need to pull the head to check for damage or retrieve the screws.
So yes, the hose with the white tie is the breather hose. The throttle shaft will move up and down a small amount. If it will move a great deal, the screws likely have come out of the throttle shaft, the plate inside the carb isn't obstructing the air flow and the engine is very well racing which a) isn't good and it will destruct itself and b) could be the reason why you are venting oil from the breather hose. The carb should come off next and have a look.