You can't just remove the balance shafts, that assembly is also the oil pump. You would have to research installing an oil pump from a 2.0L version engine but that will not help vibrations.
If you used overly cheap parts in the build that isn't going to be helpful at all.
Timing itself would trigger a check engine light if it's off at all. Balance shafts/oil pump timing won't however.
The transmission can cause issues and judging by what was wrong with the engine it's likely the transmission took damage as well. To do what he did he was racing the engine pretty hard when it came apart. Just slapping random parts together doesn't always play out well.
Parts are not cross compatible between different versions of the 2.4L which may also be part of your issues.
Flywheel, main pulley on the crankshaft are another couple of options for vibrations.
But a throaty sound when trying to rev engine along with a bad vibration sounds more like there is a misfire on one of the cylinders
Without hearing/seeing the issues it's hard to say. I would make sure you have proper compression on all cylinders, coils and plugs are good, fuel injectors are working etc.