There are a variety of ways to divvy up the political parties in our system. Here are a few that are directly relevant to the tax code, its complexity and any chance it has of being simplified.
Democrats want to do some redistribution (have the rich pay more, and then send it off to those with less income/more demands on their resources to even out the inevitable unfairness of our political system). Republicans have a horror of redistribution.
Democrats kind of like the idea of having government being all orderly and with the structure matching the function: an agency for each category of activity, regulations on a per agency basis, etc. This is complicated, so on balance, they try to simplify by moving more up to the federal level instead of doing it at the state level. But life is full of unpleasant compromises, and they'd rather make their own state(s) have the right kind of rules than wait until the feds can be convinced to do it right.
Republicans kind of like the idea of government being small, and running as much of government functionality through a common system as possible. On balance, they would prefer the federal government to do as little as possible, and anything complicated that needs to be done, they'd like to push down to the smallest unit of social organization that it possibly can. In practice, that means to the state level. The more ideological members would just as soon everything was handled at some nuclear family level. But life is full of unpleasant compromises, so if we are going to have to distribute money to people -- people with kids, people who need health insurance, etc. -- and it is going to be done at the federal level, they'd like it done as cheaply as possible, which means, tax credits. They also like it when the feds do the redistribution _FROM_ the general tax revenue _TO_ states, to do what they like with (block grants). They'd rather make their own states have the right kind of rules than wait until the feds can be convinced to do it right.
In practice, this means things like: even when everyone agrees that we should implement some kind of program, we probably can't agree or by happy with how we decide to implement that program. Democrats want the feds to send money or money-equivalents to individuals around the country who meet rule defined criteria. Republicans want states to get that money to make their own rules for who should benefit from federal largesse. You can be cynical about either or both perspectives -- the phenomenon, however, is real.
Everyone agrees, periodically, that the tax code has become unfair somehow and that it has become onerously complex and difficult to comply with. Republicans tend to focus on how this makes it hard to do business, hire employees, compete with other countries. Democrats tend to focus on how the tax code contributes to increasing inequality and that it should be made more progressive to reduce inequality. Republicans want a simple code that gets a bunch of its simplicity by having a "flat" tax -- the same percentage taken from everyone regardless of income. Democrats want a simple tax code with a steep curve after some point taking more and more of greater and greater amounts of income. Occasionally, you'll get some oddball come along and suggest a wealth tax (Piketty); in practice, outside of things like property taxes, we don't do wealth taxes in the US, and for very good reasons (administering wealth taxes is _hard_ and honestly somewhat expensive).
The two parties are unlikely to ever happily agree to the basic structure of the tax code (ignoring a true flat tax, even the number of brackets and the rates for each bracket tends to be controversial). But it is difficult to make progress towards simplification. For example, getting rid of AMT would simplify the tax code -- it would also make it a lot less progressive. Getting rid of the AMT is a pure-play Republican thing. What about the mortgage interest rate deduction? It's pretty simple to show that the mortgage interest rate deduction increases wealth inequality over time and is probably regressive. It is also one of the few items that pushes households over the line into claiming itemized deductions vs. claiming the standard deduction. With Democrats liking getting rid of a regressive and/or wealth inequality increasing thing, and Republicans wanting a simpler tax code, you would _think_ this would be a thing they could agree to change.
But only if you forget that while people vote Republican or Democrat -- quite consistently -- people who own homes and have mortgages vote at much higher rates than people who don't. And that interest deduction is kind of a big deal.
I'm not saying it will never go away. Things happen. The world changes. But the tax code retains its complexity for very good reasons.
Here is why it is likely to get worse. With Republicans in charge, with their preference for helping people via the tax code vs. creating/increasing the scope of agencies and "entitlements", if the current administration is going to make good on its various campaign promises to help people caring for children, disabled family members or aging family members (remember -- Republicans want to push that kind of task as far down the social structure as they can), it will be via tax credits. That's exactly what was promised in the last campaign season. And that will NOT lead to a simpler tax code.
I vote Democratic. That's not likely to change, at least, not until we live in a world where both parties are really, really, really clear on the right of women to decide what happens to their own bodies. Which is a world which keeps receding further, and further into the future. But I can see the appeal of delivering money through the tax code -- even while I can clearly envision a number of problems with doing so. But whatever I might think of the _merits_, I think it is fairly safe to say that the tax code does not look like it is going to get a lot simpler any time soon. Unless by simpler, you mean, even more complicated.
Oh, and I argued all that without even getting into the weeds of the costs of implementing a program with an agency vs. through the tax code.