It's a weird one for sure, borderline even, but I can understand the logic the ref must have used.
The relevant part of the rule:
(1) an attacking player, either by his positioning or by contact, impairs the goalkeeper’s ability to move freely within his crease or defend his goal; or (2) an attacking player initiates intentional or deliberate contact with a goalkeeper, inside or outside of his goal crease. Incidental contact with a goalkeeper will be permitted, and resulting goals allowed, when such contact is initiated outside of the goal crease, provided the attacking player has made a reasonable effort to avoid such contact. The rule will be enforced exclusively in accordance with the on-ice judgment of the Referee(s) but may be subject to a Coach’s Challenge (see Rule 38).
In this one, the goalie's ability to make a save is impeded by his defender getting pushed into him by an opposing player. The contact began in the crease, causing his blocker hand and stick to get levered into place, making it impossible for him to square up to the changing angle of the puck, which is why he throws his legs out and back in desperation. If he weren't being touched, he's far more likely to make that stop.
So really it comes down to the refs interpretation of the rule. Does "contact" mean direct contact between the attacking player and the goalie, or can it also mean contact caused directly by the opposing player's actions? It seems to me that the opposing player pushing the defender into his own goalie should be considered contact, otherwise what's to stop forwards from pushing defenders into their own goalies all the time? Even though it's not direct contact, it is intentional contact that impedes his ability to make the save. At least, that's my interpretation.
*edit* Just saw this addendum to that rule above, which should put it to rest:
If a defending player has been pushed, shoved, or fouled by an attacking player so as to cause the defending player to come into contact with his own goalkeeper, such contact shall be deemed contact initiated by the attacking player for purposes of this rule, and if necessary a penalty assessed to the attacking player and if a goal is scored it would be disallowed.
This play was cut and dry goaltender interference.