If there is an injury timeout and the offensive team wants the clock to continue running, or if the offensive team wants the clock to stop, the play clock will stop
Not sure where you got that idea, but it is not necessarily the offensive team that controls the situation. it is the opponent of the team that has the injury that controls the situation, which could be either the offense, or the defense. In the instance under discussion, the offense (Washington) - who were ahead in score - had the injury. Their opponent (Texas, on defense), then, had the option to accept a 10-second subtraction and the game clock would, then, be started on the referee's signal (by rule), or
decline the ten-second subtraction, and the game clock would, then, be started on the snap (by rule). Being behind in score, they (Texas) elected to decline the 10SS; thus, the game clock was started on the snap. All correct, and administered correctly on the field.
This same protocol is in place for a helmet that comes off of a player in the last one minute of the 2nd/4th periods. The
opponent of the team whose player had his/her helmet come off controls the situation.
Note that the same protocol is in place, by rule, for fouls that stop the clock in the last one minute of the 2nd/4th periods. In those cases, you can use the term "offended team" for the team that controls the situation (which, again, could be either offense or defense).
In any case, the thing that cannot happen is to decline the 10SS, but then have the game clock start on the referee's signal. That is not allowed by rule, because it is totally nonsensical.