I know that tables must have this framing structure so legs don't get wobbly,
Tables don't need this sort of framing to be stable, there are many thousands of tables (and seating benches) that don't have any — not least as seen in the pictures of the legs you purchased ;-)
In summary, how to make a stable table with hairpin legs?
- The hairpin legs need to be built well enough that they aren't too springy. The ones you link to seem like they'd be more than good enough in this regard with the third rod welded inside the 'hairpin' shape (many similar legs don't have this added rod which presumably acts as a stiffener).
- Thick top*. A thick top is stiffer and it will also provide a more secure fixing for fasteners.
- Attach the legs securely — use enough screws and the screws need to be fat enough and/or long enough to hold well in whatever material is chosen for the top (solid wood holds screws better than MDF or chipboard/particleboard).
- Use the right type of screws for the material. MDF and chipboard/particleboard benefit from using screws specifically intended for them, while solid wood works fine with normal wood screws.
- Space the legs evenly.
- Space the legs wide enough apart that they make a stable structure — the closer the legs are together the more unstable the shape formed because the table ends up top-heavy (the centre of gravity is high). When the legs are attached wider apart the centre of gravity lowers. Lower centre of gravity = more stable. If your table is not that wide for its length position the legs as near to the long edges as possible.
*If the table is wide enough. The narrower the table the thinner you want the top (within reason obviously) so that it isn't as top-heavy. 12mm material can be strong enough for a table that sees regular use, without framing or other reinforcement, although go thicker if you can.