We also don't know what's being said in the background.
Given the response of the Burnley players they must have been doing a silent check the Shaw challenge already.
How quickly is that communicated.
This would have been the VAR checking Shaw, then the AVAR must of taken over for the DOGSO or at least had eyes on it.
I don't think your theory is weightless. How directed by the ref it is I don't know.. Perhaps VAR are giving some direction.
For what it's worth I think they would have checked it in both scenarios, its the changeover of possession and one pass in between.. I just wondered if that was right or not.
As I understand the protocol and facts, there are two possible things that happened here.
Option 1:
VAR thought it was DOGSO. Once the VAR was going to recommend red for DOGSO, reviewed the build up to the OGSO and discovered the foul on Shaw, which as something that would invalidate the OGSO was reviewable simply as a foul. If this happened, the R may have agreed with everything the VAR proposed. (Though I don't think the VAR would be raising the caution in this case, just the foul that would cancel the OGSO, and the R would choose to caution based on the foul.)
Option 2:
VAR thought the Shaw foul was a send off. (If the OFR was just based on the event, it would only be reviewable if the VAR saw a missed incident failure to give red.) In that case, the R disagreed with the VAR, but since he has reviewed (based on a proper referral for the missed red), he is within his authority to give a yellow.
What seems quirky to me about option 1 is that the Shaw foul is only reviewable because of the DOGSO red--though the R apparently never looked at that play. That is interesting, as if the R disagreed that it should be red, he would not have gone on to look at the Shaw foul and the FK would have stayed and Shaw gone unpunished. I don't know that the protocols or training really address whether the R should look at the potential red that is creating the rest of the review or, where a prior foul would wipe it, are supposed to look at that first. Or perhaps it is undefined, and they just did it this way.
Definitely a case where the details of protocols affect who does or does not get sanctioned.