"only 1/3 in" - Thanks to the Americans, IDK if you mean One Third Of The Way In or "One Third of an Inch"
#Easiest way to attach a metal table frame to a solid wood table top? (image in post)
spare zenith
If it's the former, the simple answer would be "Get longer screws", but a going concern is whether or not the tabletop is going to shrink/warp after it's fixed on.
Initial recommendation will be to let the wooden tabletop sit for several weeks in the same room as the desk before you do anything with it.
MDF/Ply are great for desktops because they don't expand/contract/warp in the same way natural wood does.
Rough estimates are 1.5% movement perpendicular to the grain direction.
Which is why tabletops are usually fastened to rails with little hook things.
Screw are hella strong, you're not going to shear 8 of them moving a table.
not unless you're incredibly strong and violent.
This shows how you traditionally connect a tabletop - it's done that way so movement doesn't bite your arse.
Now - observe my nickname <- (Twas given, not chosen) - It might be over your dimension it's not worth worrying about, so at that point I will defer to avoid the Rule 7 stick.
Your existing scews are machine bolts, running into what's called a Threaded Insert.
You make a big hole, drive thiis into the big hole, now you have a smaller metal hole with a thread. Great for anything that needs to be disassembled.
If the holes in the frame have a little slack to them, so that wood movement doesn't indeed bite your arse, you could presume to undertake the same approach.
Defering again, though.
You definitely want to let anything solid wood sit in the house for a while before you commit to fixing it to anything, though - The conditions in the store and your home could be considerably different.
Mmm, they're fairly shallow, and if you drive them in with a long allen key you can keep a Square on the table surface to ensure you're mostly perpendicular
I am the Worlds Worst Driller (Seriously, I'm fucking atrocious) and I can just about manage threaded inserts
Just remember the holes in the frame aren't zero tolerance and in fact due to possible wood movement don't WANT to be, so perpendicularity isn't 100% vital