In order to have an armswing, the elbow has to clear your side. When you ask about "tucking your bowling elbow into your belly at the start of your approach", I assume that you do mean literally into your belly in front of your body. From this position it is going to be extremely difficult to get into a free armswing. With sufficient practice it might be possible to get an armswing without muscling it, but it will be difficult. It will also be difficult to have a consistent armswing. Depending on how you are setting the ball in motion, it can be a cause of getting the upper body forward, and might be a cause of at least some of the thumb problems.
If you want to anchor your elbow during your set up, rather than tucking it into your belly, I would suggest having it at the point of your hip. There are a couple of common causes of leaning forward during the push away. One is getting the push away too long. When this happens, it is very difficult (because of the weight of the ball) to let the ball swing without both muscling it, and leaning forward. The ball literally pulls the upper body forward. How to get the ball into a swing depends somewhat with the height of the ball in the setup. In all cases you want the point of elbow within the frame of the body (when looking at it from the side). This allows the body to provide a vertical support for the entire armswing. If the ball is chest high, you just let it hinge down from this position with little or no movement of the elbow. This type of armswing was more in vogue a number years ago, especially for people with slight build (e.g. women). It encourages a faster armswing, which also usually results in more foot and ball speed with less effort. (The downside of speeding everything up is that timing is a little more difficult). The other end of the spectrum on ball height, is to have it parallel to the floor; the forearm is at a right angle to the upper arm. (I would not recommend having the ball much lower than this. Any lower, and it is nearly impossible to get the ball into the backswing without muscling it.) Setting the ball from the lower position can be just hinged, but usually there is initially a short slide prior to the hinge. The slide should not be very long. Again, the elbow should not get in front of the body.
It is very difficult to diagnose and fix problems without seeing what is happening. You might think about finding a coach in the area to help. Also, while it is possible that your armswing could be causing your thumb problem, it is also possible that it could be a problem with the fit. You might want to get it checked.