hi, a few comments :
1/ R squared is a measure of _linear_ relationship between two variables. If the variables are related according to any other function, you can have a poor R squared. That's one of the advantages of using something like Spearman instead of Pearson because the former allows for any function that does not change the order of your variables. This is even more so if the type of relationship is not easily expressed in a linear equation (e.g. two variables that show peaks and troughs at similar times but are not necessarily good fits)
2/ R squared is a measure of _contemporaneous_ relationship. A chart can show a lead-lag relationship quite easily (e.g. Yt = mXt-1 + b) but if you do an R squared it would be poor unless you adjust for the lag.
3/ Before you do any statistical analysis you need to think about whether you should be analyzing levels or rates of change - I note that your charts show the level of the index and 10-year rates, but firstly most of us care about return and the relationship could be more solid on returns.

For the reasons above, I would tend to think that R squared is overly used and limits you to a very small set of relationships that rarely exist in real life. Relying on it would tend to dismiss many useful relationships and hinder exporatory data analysis.