Looks like I'll just have to trust Coinbase and figure out their screwy interface.
No, coinbase is not an alternative. There are other local wallet apps and each of them is to prefer over a web service like coin base. Before you make a fatally wrong and dangerous decision (such as giving away your bitcoins to someone else to hold them for you) you should understand and compare the risks. A sync problem in Multibit is zero risk for you, its just a minor temporary annoyance not being able to see your latest transactions immediately due to failure to download them from the network, thats what the "reset block chain" button is there to try downloading them again. Nobody can lose or has ever lost bitcoins from a sync problem, this is just not possible due to the way bitcoin works.
If you are now misinterpreting this minor user-interface glitch as a greater risk as giving your precious private keys away to a third party with remote access from all over the world and with dangerous exposure to inconsistent US regulatory agencies then this might later prove to be a huge mistake. I recommend you keep your BTC in Multibit where they are safe and controlled only by you and not share them with a whole bunch of people you have never met. Even if this means you have to click the "reset" button once every few months to force downloading the tx history if it doesn't download on its own.
I've had a number of other problems with Multi-bit, so I'm through with it. There are a couple small missing transactions that showed up for a few days but disappeared with the reset, and that has happened before. A few days ago it died and wouldn't run because of some sort of Java issue. I got it running eventually, but it now takes forever to load. This is just the straw breaking the camel's back.
I know there are other local wallets including light-weight ones and I understand their security advantages. I have some issues with being able to use them though, in the manner that I'd like. Coinbase is a very professional firm. If I can trust Fidelity with my fiat investments online, I think I can trust Coinbase with a couple BTC.