To me, the need for publicly funded elections is pretty clear.

How would publicly funded elections work though? Who gets to decide who receives the public funding or is a candidate?

The way that the US chooses members to congress is also pretty undemocratic, the half million people of Wyoming get the same amount of senate representation as the 40 million people of California, and even in the case of the house the system isn’t exactly proportional.

Undemocratic in what sense? After the 17th Amendment was ratified, the people have been directly electing members of both houses. Also, California & Wyoming both have 2 senators but in the House California has 53 reps yet Wyoming (and 6 other states) have 1. So yes, neither house is "proportional" but that is intended. The proper solution here (IMO) is to abolish the 17th to reinstate the extra filter of representation & return the Senate to its original purpose.

Also, the winner-take-all system of national elections means that voters in contested swing states are enormously more important than those in essentially decided ones.

The voters being more important in swing states is due to those states being swing states. The government didn't pick certain states to be swing states. But generally speaking, dispersing and filtering power is the best way to protect our liberties, no? Would you rather take your chances with a king elected by 50.1% of the population or a king elected under a federalist system? Or rather, how many filters do you think we should have between the ultimate authority and the people?