What is the best approach to an again and again wobbling rear wheel
Unless the cost of a new wheel that's suitable for you plus any other repairs the bike might need is greater than what the bike is worth to you, than the answer by far is get a new wheel. 

Folks in your weight range who ride a lot tend to need an especially strong and reliable aftermarket wheel. The stock advice is to get a handbuilt one. The reason for this is that a good handbuilt wheel has a number of processes done to it that make spoke breakage and loosening much less of a factor, to the point of never occurring if everything goes right. But that sort of wheel is usually around twice the price of a replacement wheel that at least has the fundamentals covered for what you need (strong rim). A conscientious mechanic installing that sort of wheel on your bike will see that you would be best served by handbuilt and be able to do some of those steps on a cheaper machine-built wheel without adding a bunch of time. (The big example being making sure the spoke tensions are high and balanced, and stress-relieving them).

The cost of handbuilt wheels can be considerable, but another way of looking at them is they're not really optional if you ride enough and are heavy. Otherwise one is liable to spend several times their cost getting the stock one fixed, as you've experienced.

Swapping the cassette on to a new wheel in and of itself and adjusting the rear derailleur if needed is easy and will work as well as it did before, more or less regardless of drivetrain wear. If you need a new drivetrain and feel it may not all be worth it, that's a separate consideration. It is true that new rear wheel time can be the moment where a bike is deemed not worth fixing.