Well, the first problem with using that board is that it doesn't come with a built-in speaker, meaning you'd either have to connect headphones or somehow shoehorn a speaker into the cartridge as well, further increasing hypothetical costs.
Secondly, unless you just wanted to use it for background music, there is no way to get the 2600 to talk to the board. It would be essentially impossible to dynamically play sound effects using this board, as the 2600 would have no way of telling the board to play sound like it can its own audio chip.
Even if you could somehow magically put the board, a battery, and some form of speaker inside a 2600 cart and it not look awful, and somehow magically convince the 2600 to address a component some 40 years newer than itself, then the third problem arises: What would you use it for?
The only kind of music that would feel appropriate to play while playing such primitive games would be some form of chiptune music, but if that's all you're going to do you probably could've accomplished that without the expensive audio components inside. Anything newer would sound out of place.
In the end, your'e no better off than when you started, and you essentially made the cartridge cost $20+ extra for no real benefit in the end.