It has actually a lot to do with it. For two main reasons:
1. All tax complications, special exceptions, special deductions, etc. usually meant to benefit some behaviors and/or punish some other behaviors. One can't wonder then that some people use these regulations to reduce their tax load - that's exactly how they are supposed to work! If you put into the law, say, deduction for making a movie, because you want more movies to be made, and then somebody makes a movie with the purpose of lowering the taxes, you got exactly what you wanted. It may be a crappy movie because its sole purpose was to avoid taxes, but that's still how it supposed to work. More you have special regulations, more you have ways to play them to reduce your taxes.
2. Tax scheming is not free. It takes a lot of inconveniences, highly paid specialists and paperwork to achieve. If the cost of compliance is high, it makes sense to scheme, but if compliance is very cheap but scheming is expensive, not a lot of people would bother with scheming. Some, of course, still will, but there is still a reverse proportion between cost of compliance and incentive to scheme.
> The problem is not simple but the root is; while the wealthy can influence the process of how much they pay in taxes, the system will continue to be in their favor.
How is it in their favor exactly? Top quintile has about half of total income in US and pays about 2/3 of overall taxes. That's some unusual definition of favorable treatment.