Lines of code is a great metric for productivity. Not only is it not broken, I would argue that's it's clearly the best. The important question to ask about this metric is "How does programmer productivity relate to value delivered to a customer?"
If you want to measure what programmers produce, lines of code added is the only metric that makes sense. Firstly, it's objective. It's easily obtained from source code repository logs. Many existing tools already can measure and track it. It can be applied to almost any programming language. Finally, because their workflows are (usually) so highly automated, they're able to measure their work product far more accurately than most. And while there are other things that programmers create (emails, documents, meeting invites, coffee stains...) by definition they write code. They are uniquely able to solve problems by writing software, so measuring the production of that software makes sense.
So if we can measure productivity this way, why doesn't it seem to be useful when we do? In manufacturing, higher rates of production almost always leads to more value. I know that all you Lean Production advocates are yelling and screaming about inventory management and muda right now, but if you were producing and selling 1000 things a day and now all the sudden you can produce 2000 of them at the same cost, there's net value there. If you can find a counterexample, please go ahead and produce those extra 1000 things and give them to me. I'll sell them on Amazon.
Software does not work like this at all. If your team was producing 1000 lines of code a day, and now all the sudden they're producing 2000, you really have no idea what impact this has on value delivered. Are you creating twice as many features? Are you refactoring to pay off technical debt? Or are you just thrashing around trying to find a design that fits your domain? Maybe you're adding features, but not ones that users need. Maybe you're just making your system harder to use.
The reason for this productivity paradox is simple: Software is not an asset. It is a liability. From a financial standpoint, creating it is much more akin to leasing office space than it is to producing finished product. It represents an ongoing cost of doing business, that may or not actually result in any value being delivered to a customer. Lines of code in a codebase are liabilities that you have to write, test, document, compile, deploy, etc....and the less of it you have for a given solution, the better.
Programmers solve problems, which can be assets. They often do so by creating software, thereby producing liabilities. In software, production is an expense. Competent programmers are able to create more value than cost when they do this, but not all programmers do. The best programmers can create value by removing code and the best solutions in software can often be achieved through simpler functionality rather than just more of it.
So there's nothing wrong with "lines of code" as a productivity metric. It's fine. You've been using it wrong. To understand the net value created by programmers, you have to look past productivity. You have to look downstream...at how and why and when people are using the solutions you create. You have to create feedback loops that extend all the way to the point where value is actually delivered, and you have to find ways to make those feedback loops fast and responsive. Otherwise, you're going to spend your time optimizing for metrics that don't matter.