Why should the American tax payer fund the building of a Russian engine in the US?
Presumably, the American tax payer would not have to fund anything other than paying for more launches.
Let me repeat, because it will be forgotten in the noise, not a single dollar of US taxpayer funds has yet been requested or been proposed by anyone involved AFAIK. There is every likelihood that this will be resolved by the market, and Aerojet/Rocketdyne and ULA and perhaps Orbital will either choose to accept the risk of Russian production, or arrange funding US production by a large block-buy order of engines.
But, in the
hypothetical case that private financing alone could not do the job, here are some reasons why the taxpayer might see it as the best use of their dollars:
-- The RD-180 is one of the highest performing kerolox engines in the world,
-- it has an extensive flight history demonstrating reliability and performance,
-- the Atlas V is already designed around it, so its use makes existing investments perform,
-- the US manufacturing base could really use the experience at building an ORSC and learning from Russian manufacturing whatever is useful,
-- the ongoing engineering experience with ORSC provides valuable new capabilities and experience and perceived credibility or TRL for follow-on engine and LV development, and
-- developing an alternative, if done by NASA/USG, would cost a billion+ and years of time, and if past history predicts will result in a cancelled program and no engine AFTER the billions and years are spent.
Are there no alternatives to the Atlas or Antares in existence?
Yes, there are. Atlas alternatives include the Delta IV, which is reportedly a little more expensive, and the Falcon 9, which is a little less proven, both of which cover only the lower end of Atlas V's capability range. To cover the higher end, there is the Delta IV Heavy, which is much more expensive, and the Falcon Heavy, which has not flown yet, and will not be ready to compete if its manifest is any guide until 2017. Atlas V is currently the most versatile launch vehicle in the American inventory, and it would be desirable to maintain that capability.
With regard to Antares, it's harder to say. Obviously it provides competition for Falcon 9 in commercial cargo, and launch assurance if a flight mishap occurs to SpaceX. It provides "industrial assurance" in the event that something happens to Elon Musk or SpaceX the company. But AFAIK they have not settled on an RD-180 re-engine to Antares, so an American production RD-180 might not even be used by Antares. (Presumably the business case for American production might force this to be settled before a production line was financed.)
Are there payloads that can only be launched on the Atlas or Antares?
For the Atlas V 541/551 launches, the only current alternative costs perhaps a hundred million dollars more. In some cases, yes, that means they can only be launched on Atlas or they won't go at all.
Not sure about MMRTG launches, whether they can go on Delta or not. Only Atlas is being human-rated for Commercial Crew vehicles; of course SpaceX has Dragon (crew) but again for mission assurance and schedule maintenance it is desirable to have two launch providers.
Are there no other engines that can go into an Atlas or Antares?
Rockets aren't Legos. Anything is possible, with a billion here and 5-6 years there, but it is unlikely the business case for an extensive redesign of Atlas would close without help from the US taxpayer. There are no close alternatives in the US, close meaning kerolox engines of roughly similar thrust and efficiency numbers.
Antares is perhaps a little different, since it may be facing a mandatory engine swap, and because Orbital may have done design work for several different engines. The RD-180 is the closest alternative to its AJ-26. Other alternatives might include restarting NK-33 production in Russia, starting Russian RD-181 production, or trying to get export approval for Russian RD-193 engines. Negotiations for those last three might be ticklish given the current situation, but who knows? Money opens doors.
I think (can you tell?

) that US production could be a good thing, and would prove to be a good investment for the US taxpayer if asked to fund a few parts of the process. SpaceX has their Merlin 1D, but unless you want the rest of the industry to close up shop and crown SpaceX as the new monopoly, there needs to be a commercially available good kerolox alternative.