As someone else has mentioned, the time-dependent nature is actually pretty important because PageRank (and there are a few others similar to this, Massey ranking is one...I can't remember them all) depends on every team playing each other team at some point and/or playing each other a certain number of times.

So, in practice, stuff like PageRank is fairly brittle, and ELO tends to work better.

I also don't think PageRank is approximation to ELO. PageRank are just eigenvectors so (I believe) rankings are proportionate to each other. This happens with ELO but to a lesser degree because you aren't necessarily looking at all the results for all other teams at all times. A lot of information is embedded into an ELO rating but you are updating match-by-match (I believe, I have done a lot of work with ELO and learned how to modify it so understand it well...I have far less experience with the matrix-based methods). So the practical advantage of ELO is actually a theoretical one, imo.

The best way to think about ELO is Bayesian updating: you start with a prior about team skills, you use this to create a forecast, and then update your ratings based on the actual result. Comparisons are Kalman filters, Markov Chains or MCMC, etc.

I will add ELO is very powerful. Yes, Glicko is better, ELO is a special case of Glicko but ELO is also far simpler than Glicko. Simple ELO models beat complex regression most of the time, it is remarkable.

Imo, it is the Bayesian-esque updating that works so well. And, if you understand what ELO is at this level, you can split this part of the model off and use it with regression or whatever you want. It is truly amazing though.