I suspect that if you use a file to enlarge the hole and counter sink the new hole, you will have a workable setup for a scroll saw table.
If your hole is 3/16 off and you try to use a larger bolt, you have a couple of problems. First, you need nearly a 5/8 inch hole to ensure centrality to the original mounting hole. Is that feasible? Next, the countersink for a 5/8 bolt is going to require a much deeper plate than the 1/4" thickness you have.
One think I suspect that you want is a very flat table. Welding messes that up by thermal warping.
Check the plate now with a ruler or (if you have it) a straightedge. Is it flat? If so, I'd just file out and countersink the filed hole. If not, you may want to find a new plate.
And the process for drilling holes in steel plate with a hand drill are
1) locate the hole with a prick punch (skinny punch, gentle hit generally for more precision, may not be necessary)
2) enlarge the hole with a center punch (harder hit)
3) drill a pilot hole with a small diameter bit. Ensure placement.
4) enlarge the hole to your desired size.
Good luck.