I am mitering some 21" boards for picture frames using a tablesaw. The first cut leaves a small off-cut to the right of the blade. Sometimes it goes flying, or even gets caught between the blade and blade throat cover. I am pushing the boards through with a mitre gauge. I cannot push the mitre gauge all the way past the blade as the off-cut then gets caught (drops into) in the slot past the blade, and prevents the mitre gauge from any more travel in its slot. Any ideas how to do this safely?
It sounds like your main problem is that you're cutting on the wrong side of a tilted blade. However, for cutting picture frame miters you should not have the blade tilted at all.
In general you should always have the work piece flat on the table if possible. This is a lot more stable (and therefore more accurate) than putting the piece up on edge. To cut "picture frame" miters you should have the blade straight, the miter gauge set to 45 degrees, and the stock flat on the table. (This usage is so common that the "miter gauge" is named after it.) This also has the advantage of allowing you to miter stock that is wider than your table saw's depth capacity.
The more typical use for tilting the blade of the table saw is in "case" miters, where you're making a box with no end grain showing. The stock should still be laying flat on the table surface, but now you'll cross cut it with the miter gauge at 90 but the blade tilted.
Any time you're tilting your blade you want the blade tilted away from the work piece. That way the small offcut rests on the table surface when it separates instead of falling onto the spinning blade. Depending on your saw's tilt direction this may mean moving your fence or miter gauge to the other side of the saw blade compared to how you typically work.