Most of the time its like Petercooper said, there are rules preventing ownership by non-lawyers. The idea being that a lawyer should not be controlled by a non-lawyer, because that person might pressure them to take actions that are not in the best interest of a client. Its not as much a privilege question (as tons of non-lawyers work in law firms) as much as it is a perceived problem of having a non-lawyer control the lawyer's paycheck (although that article does mention a public law firm would have disclosure requirements under the SEC which could conflict with attorney client privilege)..

For instance, in your example, a VC who controls a law firm will have vastly different goals than the lawyers, whose goals should be to advance the best interest of the clients.

There are investors in other industries who would not have "go public" as a condition, although they may not be VCs per-se. Law firms can be acquired for significant amounts though and they're usually very profitable, so there's an opportunity, but unlike technology/sciences/product development/retail it's really hard to weigh up the factors surrounding a law firm to establish the risk.

I don't know of any law firms that have gone public

It's pretty much unheard of (the first went public in 2007 - http://is.gd/cDks). There are lots of potential issues, such as loyalty to shareholders affecting attorney-client privilege. There are laws in the UK and US regarding this too. Don't quote me on this, but commonly non-lawyers cannot own shares in law firms in the US.