Dante clearly revered Virgil as an influence and teacher of sorts. I need help with understanding the legacy of Dante in Western literature with...

The main issues you will need to address here are what you mean by "measure up" and "a twenty-first century perspective."

There are over seven billion people in the world. The two most populous countries in the world are China and India. The most important writers in China might include Li Po, Tu Fu, and Bai Juyi. Similar in importance to Homer and Virgil in the west in India would be such traditional epics as the Ramayana, Mahabharata and Bhagavad Gita. In general, the importance of the western European literary tradition is greatest in western Europe and those parts of the world colonized by Europe (the Americas, Australia, etc.) Even within Europe, there are differences. Homer is overwhelming influential in Greece while both Dante and Virgil have strong associations with Italy. 

Comparing their legacy is a part of literary studies know as reception theory. The fields of "classical tradition" and "medievalism" focus on the respective receptions of Graeco-Roman and medieval works. All of these approaches tend to argue against thinking about how works "measure up" in some absolute manner and simply emphasize studying how the ways people think of given works change over time. 

Because Homer influenced Virgil and Virgil and Homer influenced Dante, when their works are taught in translation they are normally taught in chronological order, with treatments of later works referring back to earlier ones. In language pedagogy, though, students traditionally study Latin (and thus Virgil) before Classical Greek (and thus Homer) because of its slightly less complex syntax, making reading Virgil in Latin more common than reading Homer in Greek in schools. Far more students at the secondary level study Latin than Greek in most countries although in Greece and Slavic countries reception history is different. You might look at A Handbook to Classical Reception in Eastern and Central Europe in your university library for more details. 

One way to examine the relative influence of these authors is to search for how often their names and works are mentioned using the Google Ngram viewer which searches a huge literary corpus and reports word usage frequency. It can be found at: 

https://books.google.com/ngrams/

One problem with word counts though is that you must be careful to use author/title combinations as names such as Dante or Homer can refer to contemporary popular culture figures as well as earlier authors. 

Another possibility is looking at new editions and translations of the works. You can search a British university library union catalog quite easily at:

http://www.copac.ac.uk

A quick search will reveal Homer as twice as popular as Virgil and Dante slightly behind Virgil. This does, I think, represent a general consensus in which all three are regarded as significant poets, with Homer having the slightly greater universal appeal or influence, Virgil following Homer, and Dante being most influential in Roman Catholic educational tradition, but slightly less frequently studied than the other two elsewhere.