Death takes on the role of the Hogfather, but he also gets Susan to do the hard part of stopping Teatime's plan. Moreover, if Susan had failed it seems that Death's stint as the hogfather would have been for naught anyway, since the "old magic" would allow Teatime/the Auditors to control belief anyway.
So did Death actually accomplish anything by acting as the Hogfather?
Yes. His aim was to prevent the loss of belief which drives the magic system in the Discworld. To this end he places himself into the position of Hogfather so that children would believe in this personification and so create the disorder that the Auditors dislike:
I TOO HAVE DONE WHAT I SET OUT TO DO. I HAVE KEPT A SPACE. A MILLION CARPETS WITH SOOTY BOOTMARKS, MILLIONS OF FILLED STOCKINGS, ALL THOSE ROOFS WITH RUNNER MARKS ON THEM . . . DISBELIEF WILL FIND IT HARD GOING IN THE FACE OF THAT. ALBERT SAYS HE NEVER WANTS TO DRINK ANOTHER SHERRY FOR DAYS. THE HOGFATHER WILL HAVE SOMETHING TO COME BACK TO, AT LEAST.
If he had not retained the belief, the whole system would have been reduced to non-magical physics of the sort the Auditors like as it is regular and predictable:
“WHAT WOULD HAVE HAPPENED IF YOU HADN'T SAVED HIM?
"Yes! The sun would have risen just the same, yes?"
NO
"Oh, come on. You can't expect me to believe that. It's an astronomical fact."
THE SUN WOULD NOT HAVE RISEN.
...
"Really? Then what would have happened, pray?"
A MERE BALL OF FLAMING GAS WOULD HAVE ILLUMINATED THE WORLD.”
He got so close to not achieving the retention of belief that the Hogfather was reduced to the origins of the Hogfather myth with the ritual man and pig killed to bring back the sun at the solstice; much like Christmas is a variant on Yule, which is a Blót. Much like Christmas now largely focuses on Santa Claus (a corruption of the words Saint Nicholas) as a personification of the festival; the Hogfather is the personification of the winter solstice in the Discworld.
Belief is a strong theme in the book - look at the appearance of the various fairies (hair loss fairy), gnomes (Verruca gnome) and even gods (Bilious the Oh God of Hangovers), not to mention mythical creatures such as the "sock eater" throughout the book (see list at Discworld fandom wiki). These all appear based on faith that there is such a personification(? - is a fairy a personification?)/creature around. Socks go mysteriously missing during washing, so something must be eating them; ergo the physical manifestation of the "sock eater" with all the surplus belief washing around.
Secondarily (not related to Teatime and the Auditors), while he is filling in Death is also able to accomplish at least one thing that the real Hogfather could not have.
Death looked down at the shape under the falling snow. Then he set the lifetimer on the air and touched it with a finger. A spark flashed across.
"You ain’t really allowed to do that," said Albert, feeling wretched.
THE HOGFATHER CAN. THE HOGFATHER GIVES PRESENTS. THERE’S NO BETTER PRESENT THAN A FUTURE.
