How can I attach a metal frame with pre-drilled 3/8" holes to a hardwood table top?
If I understand correctly, your concern is that the 3/8" holes in the frame would require a pretty large diameter screw in order to have a snug fit when you screw down the table top.  Since you are dealing with Beech instead of particleboard, you want to use traditional screws instead of the pressure plugs.  If this is correct, I can offer the following thoughts:

Using the frame as a template, it is reasonable to drill 3/8" Holes in the top and use the pressure plugs that came with it.

You could buy a large 3/8 diameter screw or lag bolt.  Just be sure to pre-drill you pilot hole large enough so you don't crack the top.

You can drill pilot holes in the top and screw it down with bugle head wood screws. (AKA flat head screws, AKA flush head screws, AKA countersink screws)  The cone shape of the head of the screw will center the screw in the hole and prevent the smaller shaft of the screw from wiggling around inside the larger diameter hole.

You could cut off the bottom part of the plastic pressure plug so that the length is the same as the thickness of the frame.  You could then insert the plug in the frame, drill your pilot holes, and use the plug as a spacer inside the 3/8" hole to keep you screws from wiggling around.

In any case, be sure you use a good quality, high strength screw.  A cheap screw (drywall screws especially) may snap off when you are tightening them in beech.

Hope that is helpful, good luck with your project!