Working legs and core may, if that is all you do, and if that indeed allows for greater velocity, may work the other way by allowing greater arm speed and putting more stress on the elbow.
There are, I believe, 10 distinct muscles that cross through and support the ucl. Strengthening these muscles requires some very specific work, though. The more developed these muscles, the more support they provide the ucl and the more arm speed the elbow can withstand. Some of these wouldn't even at first glance appear to effect the elbow. I'll give you an example. I (and others) have noticed that pitchers will tend to have a tightness problem in the forearm for a time before the ucl gives and they need TJ surgery. This precursor, however, seems to go largely unnoticed in MLB. I often see a pitcher go on the DL with a "forearm strain" only to come off the DL and the next injury is the elbow. There seems to be some correlation between muscles in the forearm giving out and future ucl problems, perhaps from a decreased amount of muscular support.