You called it the 'brake' which is correct. When you release the engine/blade control bail, a strong spring forces a brake shoe against the engine flywheel. At the same time, a switch cuts off the ignition. The brake is there to stop the engine/blade as soon as possible. If you are operating the mower with the bail released and the ignition cut-off disabled, what remains of the engine brake shoe is trying to stop the engine, which is powerful enough to overcome it. This puts a load on the engine as if it were in thick grass, which opens the throttle, causing the engine to go into 'power mode' which results in the smooth running 'power hum'.
When you engage the engine/blade control bail as in normal operation, the load is removed and the engine returns to 'idle mode' in which surging is normal. It should return to 'power mode' when mowing presents enough of a load to slow down the engine and let the air vane open the throttle.
So, yes, you are messing up by not using the bail when the engine is running because the brake assembly can cause excessive wear on the flywheel, and having the ignition switch disabled is unsafe.
It's possible that during all the work, the air vane somehow became jammed to some extent and is no longer free enough. Remove the air filter with the engine off and see if the throttle plate is in the wide-open position, which is correct. Push the air vane over about halfway and release it - it should move freely and snap back into position.
A 'rich' condition cab cover up many problems, which is well known in auto mechanics, and the same holds true for the Dura Force engine. You can over-jet it to force it to have a smooth, 'burbling' idle, but in fact there is no need for that. It's obvious that it responds to a load by running in 'power mode', so it's doing what it should. Otherwise, possibly the new jets have gotten clogged to some extent over the last few years of use.

Wow, what an excellent and prompt response!!! Thanks so much!

I'll be checking the vane and reporting back. Will definitely stop "driving while braking" from now on... however, now I'm wondering, when I reattach the ignition cutoff wire, it surges (thanks to your explanation I understand why: the engine is not under significant load)... but it still surges except under the very very thickest grass. It surges far worse than even before the rejet. I have a Bahia grass which is not very thick nor dense when cut at usual heights. After the rejet this mower was just absolutely a dream for two seasons, and these probs didn't start till this year when I had the coil then recoil and cable issues. I was somehow concerned that a faulty coil might cause some of this craziness, or that the brake cable not travelling properly would cause some of this.
But now I'm thinking that after checking the vane a nice cleaning of the jets would probably be good (the filter I installed was at the end of the last "good" season) so maybe it is gunked up in there.
Is Seafoam a good idea or should I just take it all apart and have at it with carb cleaner?

Yeah, the gas filter might be clogged up. Otherwise you'll probably have to clean the carb/jets.

Thanks for the post. I think it matters a lot. Not mowing the lawn can result in heavy fines, even jail time.