If you not change the construction so the wood is able to move everything you do is in vain. It will crack again.Not sure how the gaps emerged, as I say its definitely pre-WW2. The planks are spine jointed then screwed to the base from underneath. The planks arn't able to move freely. It would be a monumental job to take it apart and redo the spine joints and then reattach everything. Its beyond me, with my hand tools and hand planers. The table is about 10 ft long.
I would take the top of, take a circular saw and a guide ruler and allong that cut right at the seems.
That way you get an even edge and then glue with a foodsafe woodglue, like titebond 3, which has some filling capacity.
After the glue-up re-attach the top with a moving construction, type depending on form of the base
