I dont think two-party systems necessarily are undemocratic, but I'd say they tend to be less transparent as any negotiation and compromise will be handled internally within the parties, and you as a voter never really knows which facet of the party your vote will strengthen.
I don't know if it actually works that way. If you look at Italy, for example, it's so fractured by many parties, that no legislation gets through, and it just becomes a political circus. No one agrees on anything, no majority whatsoever. Of course, that may be an extreme example…
First Italy is not your average multi-party system as it AFAIK has a low bar (2%?) for entry. Second the fragmentation and instability is somewhat exaggerated. On paper they've had a lot of government changes, but most of those have been reshuffles within a ruling coalition. Actual crisis and deadlock is rare. Italian governments have been more stable than Norwegian (that country well known for political anarchy and bickering) ones during the last few decades. The country has serious issues, but they dont come from the multi-party system.
There are also at any rate a ton of pretty technical solutions that can be applied to make governments in multi-party systems more or less stable.
Well, we have gridlock right now that seems insurmountable. If everybody had to get votes from someone else to advance a piece of legislation, it would have to reflect more than just one party's values, and you'd have alliances and discussions instead of fullscale rivalry and war. Though of course it could get carried to extremes if you had too many parties.
US gridlock comes from the checks and balances and the executive and the legislature having separate mandates. It can be impractical at times but works as the founding fathers intended, and is in some sense more democratic than the parliamentary systems most other countries have. The US system is from what I can tell designed to prevent concentration of power (partially due to the diversity of the country, partially due to the discouraging example of George III's attempts at increasing royal power before he went crazy). And as I've said before our prime ministers in many ways have much greater power than US presidents, even if they formally are at the mercy of the parliament and the pres isnt...
I think the US will want to hold on to checks and balances and such for a number of reasons. With that in mind I think it would be a good idea to reform the election system along these lines (which would give you a multi-party system without introducing Euro style parliamentarism):
The house: Change to proportional representation for every state. This would reduce the problem of gerrymandering and the number of "wasted" votes from people living in the wrong district. You can very well have personal mandates within a proportional system (Finland has one solution to this problem where you tick your preferred candidate of a party list) so people can have their own congressman.
President: Direct proportional election. If no candidate breaks 50% the two frontrunners go into a second round. This way it'd be worth it for democrats in Texas and Republicans in California to vote. The arguments for the electoral college are somewhat obsolete, and when you vote for
one individual parts of the country will feel short changed no matter what.
Senate: The same system as in the presidential election, but for each state.