How to state that you “rigorified” an existing paper without sounding arrogant or condescending?
It sounds to me like you are viewing the older paper through the lens of the type of paper you would have written in the first place and are finding it wanting.  It also sounds like you are viewing your own work as "fixing the flaws" of the older paper, almost as if you want to replace the older paper in your mind with your paper and pretend it had been written that way all along.  More quantitatively you have your eye on the mark that the older paper should have hit, you are filling in the gap between the actual old paper and your eyeballed mark, and since you are measuring from the eyeballed mark you are giving the difference a minus sign.  This framing seems to be behind most of your problems.  Try recasting the entire thing more positively in your own mind.

1) Someone else published an inspirational paper way back when.  This paper introduced some formalism and concentrated on its application to a physical problem.

2) Your paper gives much deeper attention to the mathematical aspects of the formalism, while also extending the formalism.  Since 1) showed that the formalism is interesting and useful, your work has evident value.  

There are no "minus signs" in the above description: 1) + 2) = your eyeball mark.  

Unless there are actual mathematical errors in the older paper, you don't need to say that you are fixing or "rigorifying" (not really a word, by the way) the older paper.  By the way, I don't know whether you've heard my rant about this use of the word "rigor" in mathematics and its applications.  As an adult mathematician I have become increasingly skeptical of mathematical "rigor": the other paper either made mistakes (which you will need to correct), made claims which were unjustified or insufficiently justified (in which case you make clear that what you are contributing is the justification and not the claims themselves) or they weren't doing mathematics at all (which is fine: that's what you're adding).  Adding rigor must mean one of the things above, right?  It is not some generic ingredient that you can sprinkle more or less liberally over a piece of "unrigorous mathematics" and make it rigorous.  

In summary: unless there are some clear mistakes, you don't need to bill your work as fixing their mathematics.  You can bill it either as adding the math or adding more math.  Both of these are good things added to other good things: no problem.