A characteristic of many endgames in chess where the result is clear is that pieces leave the board quickly to make the eventual win obvious. What we have seen with the resignations of some (Davis, Baker and Johnson at the time of writing) is but the first stage in that process. As I had anticipated, the Brexiters have split, probably for two reasons. The first may involve calculating what the best way of becoming May’s successor is (remember any calculation does not need to be correct). The second is about whether trying to bring May down is more dangerous to Brexit than accepting defeat and playing a very long game. Let me expand on this last point.
BINO (Brexit in name only, or something very close to it) is not a stable position in the long run for a large country like the UK. The very long game for Brexiters sees BINO as a first stage in a gradual distancing of the UK from the EU. The big problem with this strategy is demographic: Brexit was a vote of the old against the not so old. For that reason the instability of BINO is more likely to lead to rejoining, although when that happens depends in part on the EU. But bringing May down could simply backfire and halt or even end Brexit, as it can only be done by enough Brexiters joining Labour in voting against May’s Autumn deal with the EU. If the deal is voted down by parliament we are in ‘anything can happen’ territory, and that anything includes leaving with no deal, no Brexit at all, a new Prime Minister or even a new government. Remember also that it is easy for Brexiters to make threats now, but actually voting against a deal is something else. 
A vote of no confidence in May among Conservative MPs, although it will probably be talked about endlessly by the media, is also the least important event in all this. May will win, because Remainers do not want to risk a Brexiter Prime Minister. If Brexiters have any sense they will leave any vote until later anyway, for the reasons I will now explain.
We have seen the first stage of the Brexit endgame, the first set of pieces to come off. The plan hammered out at Chequers, as Chris Grey explains, is the basis for negotiation that I said May needed to get on the table, but it also cannot be accepted by the EU for many reasons. In terms of pieces still to go, the elaborate attempt in the Chequers plan to give Fox the appearance of still having a job will not stand. But having to concede in effect staying in the Customs Union, and seeing Fox (and others) go? is the least of the three changes the EU will probably require compared to that plan. The EU is unlikely to accept a goods only Single Market deal for the whole of the UK, and is even more unlikely to accept the UK staying in the Single Market without also having freedom of movement.
How the endgame is played by the EU now becomes important, because it will determine when (not how) this all ends. The end result for Brexit if it happens - some form of BINO - is not in question unless something very surprising happens within the EU. But May will be desperate to avoid that becoming clear for as long as possible, and it is up to the EU the extent to which they let her play that game. To continue the chess analogy (my apologies for those who do not play), May wants an endgame that allows her to postpone the inevitable (i.e. to avoid agreeing to BINO) until the time control period of 40 moves comes to an end (we leave in March 2019, and enter transition). That requires a withdrawal/future framework deal that is vague enough that it is not obvious that freedom of movement will continue. Even that may not be enough for May to get any deal through parliament, but that is all she has to play for.
The calculation the EU now has to do involves working out what outcome is most likely in ‘anything can happen’ territory. If they think there is any risk of no deal they might play along with May’s attempts at fudge (although the Irish backstop cannot be fudged). But if they see as the most likely outcome that Brexit comes crashing to an end, then maybe they will play for that form of endgame. That calculation will have a strong influence on how this all ends. Which is not surprising. By triggering Article 50 the UK government gave up control and put our fate in the EU’s hands, like a novice playing a grandmaster.
 Actually ending a government is much harder than most imagine after the Fixed Term Parliaments Act. Would Brexiters really join the opposition parties in a vote of No Confidence in the government and become responsible for a possible/likely Corbyn led government? At the moment a vote against any deal with the EU would lead to a No Deal Brexit, but I suspect May has enough leeway to stop that (and the EU would always accept a request for more time to avoid that outcome). She could even threaten the Brexiters with ending Brexit if any deal falls: unlikely I know but it would put various cats among pigeons.