Does the sun have nonzero electric charge? The sun is a star, meaning it's a ball of gas undergoing nuclear fusion in its core. The sun, like any other star, experiences line-driven stellar winds due to the transport of ions from its interior which, as I understand it, are amplified by the magnetic field of the sun (since the magnetic fields provide a longer lever arm to carry away the ions that compose winds). 

Does the sun have an overall nonzero net electric charge?

In such a dynamic system, I would expect that the sun would have a nonzero net electric charge since the winds are carrying away charged particles so there is a net out-going of charge. 

If the sun does have a net charge, are there processes that work to
  return its net charge to zero?

Or am I simply wrong in thinking that the particles that compose the wind are charged themselves? Are they themselves neutral particles (afterall, they are atoms)? If so, are there other reasons why the sun would have a net charge?
I've read this post, but found the only answer unsatisfying since it relies on the idea that electrons are "easier" to transport via stellar winds than protons, but we're talking about heavy nuclei (i.e. carbon, nitrogen, oxygen) that have more momentum than electrons for a given wind velocity.
 A: This 

In such a dynamic system,

encodes a pretty big misconception. The Sun is a dynamical system, but it is in a dynamical equilibrium and it will remain in that dynamical equilibrium until it runs out of hydrogen to burn. This means that this

the winds are carrying away charged particles so there is a net out-going of charge

is impossible. If there is a net out-going of charge, that would mean that the Sun has a constantly increasing rate of charge, and that is not sustainable in an equilibrium system.
It is correct to say that the solar wind carries away charged particles, but the requirement of dynamical equilibrium then dictates that the amount of positive charge carried away by the winds must be precisely equal to the amount of negative charge carried away by the solar wind. (Potentially, there could be some small oscillations ─ but on average, which might be over a few days or over a few years, the total positive and negative currents need to be exactly equal.)
All of that tells you things about the rate of change of the charge, but it doesn't tell you about its absolute value, and this is where the other argument that you referenced comes in:

but found the only answer unsatisfying since it relies on the idea that electrons are "easier" to transport via stellar winds than protons.

You'd better start digesting that one better, because that really is the answer: electrons are easier to transport via stellar winds than protons, because of their lighter mass, which means that they are more readily removed by thermal effects. (This is analogous to how lighter molecules escape more easily from the Earth's atmosphere.)
However, as mentioned above, the net rate of escape of electrons needs to be exactly equal to that of positive ions, or we'd have an ongoing accumulation of charge that's incompatible with the static situation of a dynamical equilibrium. Thus, there needs to be some effect that counteracts these thermal effects, and which makes it harder for electrons to escape.
And that additional effect is, of course, the net charge of the Sun: the Sun is positively charged, which means that there is an additional force on electrons that attracts them to the Sun (while at the same time repelling the positive ions). In the dynamical-equilibrium level, this additional bias is exactly enough to counteract the difference in the ease with which thermal effects remove electrons vs ions.
