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I'm interested in knowing why Palpatine sent Anakin to the Jedi temple during the Jedi purge, rather than straight to Mustafar to end the war.
Would the Jedi at the Temple been able to defeat the clones or would the sheer number of clones overwhelmed the Jedi regardless? Was Palpatine worried that the Jedi at the Temple might win if he didn't send him?
Is there any specific reason (in-canon) to describe why Anakin was sent to the Temple first?
The junior novelisation indicates that Palpatine felt very confident that he'd sent enough troops (almost a thousand) to pacify the Jedi Temple. His decision to send Anakin seems to be far more focused on securing his loyalty (by pushing him deeper into the Dark Side of the Force) than for any specific military benefit.
Note that aside from killing a few kids (and dispatching Shaak Ti in the original script) his involvement is pretty darn peripheral:
Darth Sidious — Chancellor Palpatine — stood alone in his enormous office. He’d sent his new apprentice and a battalion of clone troops to the Jedi Temple. That would take care of the Jedi on Coruscant. He scowled slightly under his hood. He sensed that his apprentice was not yet as fully committed to the dark side as he should be. Well, destroying the Jedi here should certainly tie Anakin closer to his Sith identity — Darth Vader.
For the record, the Clone Troopers seem to be doing pretty well against the Jedi even without Anakin's assistance. Now, admittedly it takes ten of them to kill one Jedi, but sheer weight of numbers is going to be sufficient to take out all but the most battle-hardened of Jedi, and those were the ones leading troops offworld.
Apart from Valorum's answer, there is nothing in the canon that I could find which would indicate that Vader was absolutely necessary. The purpose of Knightfall was a) to kill the younglings and any remaining Jedi on Coruscant, b) to gain access to the Archives and the Holocron Chamber, and c) to reprogram the Jedi beacon to call any survivors of Order 66 home to be ambushed. While Vader certainly helped, there is no indication that:
a) the clones couldn't eventually overcome Cin Drallig/Shaak-ti/the Temple Security Force, especially as Sidious sent clones from the elite 501st that had been specially trained to fight against Jedi (ROTS novelisation by Matthew Stover)
b) Vader was necessary to break into the Holocron Chamber or the Archives - Cad Bane did it in TCW 2x1, with a little help from the inside (Cato Parasitti imitating Ord Enisence and then Jocasta Nu) plus plans provided by Sidious, and he did it when a lot more/stronger Jedi were around
c) Vader/a Force-sensitive was necessary to reprogram the Jedi beacon to transmit the Return-to-Coruscant message to recall the Jedi to the Temple. The beacon was located in the Jedi Temple central security station in the communication center at the base of the High Council Tower, and Bane got his droid TODO 360 to break in there during his theft of a holocron to open the Kyber memory crystal; plus, during Knightfall, it was actually the clones that took the center while Vader was directing the operation from elsewhere. While a Jedi could more subtly control the beacon the same novel mentions that the clones themselves could reprogram it.
In reply to phantom42's point about attrition, the description of the battle in the Knightfall article on Wookieepedia indicates that while other clones were going around mopping up, Vader was leading a sortie to secure the communications center and the Archives. From this I think you can infer, rather than opine, that these were primary goals for the Emperor. A long drawn out battle would have given the Jedi more time to destroy the Archives and either steal and escape with or destroy the holocrons, plus it would have reduced the efficacy of the Jedi beacon trick (and as later events showed (Kento/Mallie Marek, Rahm Kota, Yoda, Kenobi et cetera), the more Jedi that escaped the more problems the Empire had). This I think would be the strongest indication of why Vader was there, but it is never explicitly stated in the canon that I could find.
In response to OP's point about the efficacy of the clones vs the Jedi and Vader's role in it, this is an example taken from https://starwars.fandom.com/wiki/Cin_Drallig#Order_66
Though Jedi Masters Shaak Ti and Jurokk directed the Temple's defense the night of Order 66, Drallig nonetheless played a major part that fateful night. He was in the process of giving lightsaber lessons to teenage Padawans Bene and Whie Malreaux when the Sith Lord began his murderous rampage. Deep within the Jedi Temple he prepared for Vader's coming, directing such Jedi as Olana Chion in the Temple's defense. Engaging the clones alongside Serra Keto, Drallig’s efforts proved initially successful, as he managed to penetrate the occupying forces' upper level defenses. As Vader was alerted to this, however, he went on a hunt for his former lightsaber instructor. Entering Drallig’s territory, Vader discovered what carnage had been wrought there, with dead clones and the signs of battle being everywhere. Before Drallig could move against the Dark Lord, however, Serra Keto confronted Vader on her own, and was slain.
As Drallig rushed into the room, he witnessed Keto fall to the Sith Lord. Engaging Vader in the Temple training hall along with two of his students, Bene and Whie Malreaux, they were nonetheless unable to defeat the Dark Lord. Vader outclassed both Padawans, choking Bene with one hand while he dueled Drallig with the other. The battlemaster himself ultimately proved unable to defeat a raging Sith Lord in combat. After the demise of the Padawans, Vader cut down Drallig following a short exchange of lightsaber strikes. The incident was recorded by Temple security camera TR4-121.
Other than that, there are only circumstantial quotes indicating how the battle progressed and what Vader helped with, but you can't objectively separate Vader's role from those of the clones since they worked in tandem.
