That is a lovely idea. Unfortunately, a scoring scheme has one foot in the indexing process (that thing that the google bot does) and another in the querying part, so switching schemes would often mean you would need to re-index your data to cater for the new metrics you now need for a new type of scoring.
Neither indexing nor querying does the ranking. Ranking is done after indexing and can be either tf-idf , pagerank or combination of that.
Once the document similarity to the query is calculated, by for example vector space model, the documents are ranked by pagerank.
What OP is saying that instead of pagerank we can have other ranking methods which is surely plausible.
Sure, but what I was saying was that what good is a new ranking method, when you only have at your disposal the same set of metrics as the method you are trying to replace? A new ranking would quite often mean adding new metrics. For example, when Lucene when from tf-idf to bm25 they added lots of new metrics to be able to cater for the new algorithm.
