Iterating on my MVP for a year, got to know my users, ready for open beta, but web app is clunky. Does it pay to trash and start over?

It doesn't sound like much of an MVP if you've been focusing on product features rather than on clients.

If I were you, I'd clean the product up, leaving it a real minimum functionality, and I'd start working on getting clients to try it, and getting their feedback before doing any redesigns or anything at all. Learn from your clients, see if they find value in what you do, and then hire a good developer, get a technical co-founder, or get a good programming agency that does exactly what you need, and not a clunky thing full of needless features.

If you started fresh do you have a clear idea what the core features would be to really get the traction? If you do then go ahead an pair down the features and start a new version. There's a well known book called the Mythical Man Month where the author advocates throwing away your version 1 since its mainly used to get to product market fit - and in your case since its all messed up, I'd advocate it.

Two things.

If the market is big enough and early enough that your 'product' can serve as a concept demo, then clean it up and use it that way for early investors.
Understand and act on the inherent limitations of the whole 'MVP' concept. In the 20-something led startup world the MVP has become synonymous with the term 'shit sandwich'.

Iterative development is not an excuse to waste a customers time on debugging your product nor is it a rationalization for treating customers or early users like guinea pigs in a lab experiment. It's a con that has been validated by VCs and incubators as some kind of junk science for their own reasons, i.e. identify early markets to 'bank on' and placing multiple bets on those markets knowing full well most startups will fail.

The basic answer is to stop being a pack follower and begin filtering the BS from so-called thought leaders and revert back to proven best practices in product management.