There are a number of points in your question that I will address separately.
When advantage is played on a DOGSO situation, and a goal is scored, should the offender always be cautioned?
From what I can gather, the first question is whether attempted but unsuccessful denial of a goal or denial of an obvious goalscoring opportunity by committing an offence should always result in a caution for unsporting behaviour, or whether the referee can choose to not caution the player.
From Law 12, Section 3, Laws of the Game:
If the referee plays the advantage for an offence for which a caution / send off would have been issued had play been stopped, this caution / send off must be issued when the ball is next out of play, except when the denial of an obvious goal-scoring opportunity results in a goal the player is cautioned for unsporting behaviour.
So if the player would have been sent-off for denying an obvious goalscoring opportunity or goal had play been stopped, the send-off is instead downgraded to a caution for unsporting behaviour. By the letter of the law, this caution is mandatory.
In previous editions of the Laws, this caution was discretionary (notice the wording change - must vs may).
p. 132 of the 2015/16 FIFA Laws of the Game:
If the referee applies advantage during an obvious goalscoring opportunity and a goal is scored directly, despite the opponent’s handling the ball or fouling an opponent, the player cannot be sent off but he may still be cautioned.
It will be interesting to see whether this was an intentional change or an oversight - there have already been a number of errors in this edition of the Laws, as it has been a significant revision and was undertaken in a very short period of time. Some have been amended in the FAQ (eg. see Q6 and Q7 here - reading the Laws without the FAQ initially implied a direct free kick rather than an indirect free kick restart for dissent) and others have not (as shown in this answer).
This is where I am questioning the referee's decision. It should not matter if the attacking team ended up scoring anyway: a goal has still been prevented, because what was scored next was another, different goal. So I think the right decision should have been to let the goal stand, and send the defending player off instead of just cautioning him.
The second element, is whether the player should be sent-off, for example, if they handle the ball on the line, the ball bounces outwards and then another shot is taken - ie. the first goal is denied illegally, but the second one is scored.
Returning to the quote from Law 12, Section 3, Laws of the Game:
... except when the denial of an obvious goal-scoring opportunity results in a goal the player is cautioned for unsporting behaviour
I believe there has been editorial oversight here - how it should have read is:
... except when the denial of a goal or an obvious goal-scoring opportunity results in a goal the player is cautioned for unsporting behaviour
The reason the send-off is impossible if the goal is scored, is due to advantage clause in Law 5, Section 3 of the Laws of the Game:
- allows play to continue when an infringement or offence occurs and the non-offending team will benefit from the advantage and penalises the infringement or offence if the anticipated advantage does not ensue at that time or within a few seconds
When a player commits an offence that denies a goal or an obvious goalscoring opportunity, there are only two possible outcomes:
- the denial results in a goal, meaning the anticipated advantage has ensued. The player is cautioned for unsporting behaviour, and a kick-off is taken by the conceding team.
- the denial does not result in a goal, meaning the anticipated advantage has not ensued. Therefore, the referee must award a free kick or penalty kick, and sends-off the player for denial of a goal or goalscoring opportunity (unless the offence occurred in the penalty area, is a genuine attempt for the ball and was not pushing or holding, in which case the player is cautioned for unsporting behaviour - see Law 12, Section 3, Denying a goal or an obvious goal-scoring opportunity ).
The sending-off punishment is mostly restorative in nature. If the denial results in a goal, even if it isn't the original attempt, there is no need to give the scoring team the advantage of an extra player. That is why the offence results in a caution rather than a send-off. The caution is given to deter the player from committing further tactical fouls in future.
In the situation you linked, it has been noted in other answers that the player actually handled the ball after it had crossed the line. The reason the player was cautioned was because the referee believed that the player deliberately handled the ball while it was in play, and had advantage not ensued (ie. by a goal being scored), he would have been sent-off.
However, since the player could not possibly have been sent-off here, since the ball was out of play when the deliberate handling occurred, the referee would have been correct to not caution the player. A verbal warning would have still been appropriate.