It actually don't make sense that the defender have a chance to break the grapple on the attackers turn.
Unless the attacker choose to gamble for a better grip.
By my understanding breaking the grapple is what the defender is trying to do during his or her turn.
I guess this is the rub, then. Especially a fundamental disagreement on the passage I bolded.
In 6e, it's demonstrable (this isn't opinion, this isn't up for debate) that if the grappler wants to, quote, "simply opt to keep holding the defender" that is a Restrain action. It costs a major action during the grappler's turn. And the grappler can fail this test. Consequences of failure isn't said explicitly, and if you say implicitly that means "the escape threshold that was never mentioned doesn't get any bigger" I'm gonna hard disagree and insist the more "obvious" implicit consequence of failing a test to sustain a hold means you fail to sustain a hold.
Often times you have to look at 5e to discern what 6e is trying to do, but with respect this isn't one of those times. The rules for grappling are (with the exception of what does the Break Free action roll against) clear AND a break from how 5e did it.
The flowchart for 6e goes as follows:
(Grappler's turn)
1) Must first succeed on a Close Combat + Agility vs standard defense test. (Grapple rules, 2nd clause of 2nd sentence, pg 111).
2) Then instead of the defender making a soak test, the attacker makes another custom test: STR + net hits vs STR. (Grapple rules, 3rd sentence). Success on this custom test has a binary result: either the target is grappled, or the target is not grappled. There is no mention of net hits on this test having any importance at all, much less tracking them to establish a threshold for future tests by the victim. (Grapple rules, 4th sentence).
(victim's turn)
Assuming the target has been grappled, the target now suffers the effects of BEING grappled (sentences 5 thru 7).
3) the victim can take any actions allowable by the grappled state, including the special one, Break Free, called out below the special actions allowed to a character who holds a grapple.
4) if the victim tries to break free, you first evaluate the AR vs the Defender's, and the victim rolls STR+Close combat. Exactly what against is not called out, and we disagree about whether it's more "sensibly" a threshold that was never established or STR+Close Combat, as almost everything else in 6e grappling uses (see below). Failure on this test, or failing to even attempt this test, results in not breaking free from the grapple.
(Grappler's successive turn)
Assuming the defender hasn't broken free, the grappler has 4 choices:
1) attempt to simply maintain the status quo. This is represented by the Retain action (STR+Close combat vs STR+Close Combat)
2) attempt to harm the defender. This is represented by the Damage the Opponent action, and is adjudicated differently than a regular attack. Of note, the test itself is STR+Close Combat vs STR+Close combat (not including some penalties being applied to the defender due to being afflicted by the grappled state)
3) attempt to throw the defender to the ground. This is represented by the Tackle the Opponent action, and is also adjudicated differently than a regular attack incorporating a called shot to knockdown. This action ALSO is resolved via STR + Close Combat vs STR + Close Combat (again, not including grappled penalties for the defender)
4) do something other than keeping the grapple going. Obviously, this should end the grapple immediately.
I can only see 2 ambiguities in the 6e grapple rules:
A) do Damage the Opponent and Tackle the Opponent also include maintaining the grapple? For the reasons I gave in earlier posts upthread, I believe the best answer is yes. Certainly superior to " you have to spend 2 major actions to hold AND hurt or throwdown."
B) What does Break Free roll against? Again, since thresholds are never established, tracked, or built upon... and everything else goes off STR+Close Combat vs STR+Close Combat, I see little reason to not presume the defending half of this opposed test was inadvertently omitted... certainly a better view than presuming the inadvertent omission was forgetting to establish the existence of a threshold as well as a mechanic to build upon it, as you envision.