This question needs to be taken from a couple different angles. What was Alaric's goal in sacking Rome? What would have been the Imperial reaction if he did? And finally why didn't Alaric either declare himself Emperor or at the very least remove the existing political authority.

The Goths had been a presence within and around the Empire for well over a century by 410 CE. Within that time they had gone from foreign people to a integral part of the Roman military apparatus. The majority of the Emperor's by this point in time had a key barbarian general or half barbarian emperor who at one point or another would serve as Magister Militum. Quite a few time these generals were Gothic. Also the Gothic warriors served as a significant component of the auxiliary forces which we refer to as the Foederati. This has become the status quo for generations by 410 CE. So it is not hard to imagine that the Gothic people within the Empire considered themselves to be a part of the Empire, as opposed to seeing themselves as foreign Invaders.

Alaric himself had for a number of years, on and off, held official military posts as appointed by either the Western or Eastern Imperial offices. When he had marches Italy, his intent had been to secure his post and recognition as an Imperial Officer of high ranking. His people had already held lands in what is now the Balkans for many years. This land is the frontier lands and also holds the border between east and west. Being located in this area the Goths were often apart of the chess game between the west and the east. Most recently they had been fighting against the west who was led by the de facto ruler Stilicho. But after his death there was a lot of chaos and Alaric hoped to capitalize on this by securing an Imperial posting. This does not necessarily mean that he was looking for new lands, though we know that was the end result. As you've pointed out, Rome was not the capital and had not been so for a very long time. It's remaining value was in the symbolic value and the wealth of the citizens. But it's political value and been shown to be negligible back in the days of Constantine.

Given this context, it's highly unlikely that Alaric would have wanted to seize the city and hold it. It would make holding an Imperial post impossible as having captured the symbolic city would have been insulting to the Imperial office and they would not have been able to justify giving a post. Also the city had long lost it's political potential making it a bad capital for the Goths to hold. It would not allow them to control food, taxes or recruitment and thus would leave them in a weaker position than they held along the Danubian frontier. The land that they would eventually hold in modern southern France is a better position. It has wide open spaces for agriculture and also lies on the major trade lane between Ravenna and the majority of the Imperial holdings. Rome was no longer the city that it once was.

Now to focus mostly on the second portion of your question, why didn't Alaric seize control of the Empire? This is a little tough to say as it's all hypothetical. Alaric died during his invasion of Italy, so it is possible that he would have attempted to seize control of the Empire. I personally find it unlikely though.

For starters, none of the barbarian generals at this point had seized control of imperial office. Their highest ambitions were to serve as the de facto head of state. For Alaric to call himself Emperor would have been a dangerous proposition that would more likely have endangered his people rather than empowering them. It would have created potential enemies. Though Honorius was not exactly the most loved Emperor, for a Gothic warlord to declare himself Emperor would most likely have unified the generals in Africa, Iberia and Gaul against the Goths. Also any barbarian Foederati who were not Gothic would likely percieve the power grab as a threat. There is also the threat of the East. When Odoacer eventually seized control of Italy, he did so with implied approval of the Eastern Imperial office. He was not approved to become an Emperor but to rule the west as Rex, paying all due homage to the Eastern Emperor. By Alaric calling himself Emperor he would have been declaring himself a threat to the East.

Alaric also had no claim to legitimacy. Whereas Honorius was born in the purple, Alaric was a general with no imperial connection to speak of. If he was to declare himself Emperor he would have to do so without any support whatsoever.

Though as you pointed out, he would not necessarily have to call himself Emperor, he could have gone for de facto ruler similar to Stilicho had been. But that being said he lacked real authority to do so. Yes he controlled a sizable army and had captured Rome. But the political power of the Western Empire was not in Rome. Western political authority was through being able to hold influence over the Imperial office, which he could not hope to gain while Honorius sat safely behind the marshes and walls of Ravenna. Also, it was through controlling the support of the provincial generals, who Alaric could not count on. Alaric lacked the ability to ensure control over the west. If he tried to seize it, it would have to be a gamble whether he would achieve it and he would have been betting the lives of his people. Perhaps if he had lived, Alaric would have taken this bet, but I find more likely that he would have continued on his quest of obtaining Imperial recognition and then following a similar path that his brother in-law took.

Please let me know if you have any questions about my answer or if you have any follow-up. Thanks!