strangely a lot of books talk about $\lambda$ calculus without mentioning Lisp or Scheme, modern programming languages based on it, leaving students unfortunately with the idea that its old and abstract and mostly theoretical. studying Lisp or Scheme can be a great angle to immensely help understand $\lambda$ calculus.
If it is just that, why is it advantageous to use λ-calculus over regular function notations to calculate algorithm run time?
there are many advantages to using Lisp or functional programming and calculating algorithm run time is just one possibility (although it would be helpful if you cited a ref for that). since its already in functional notation sometimes determining the formulas for run time via induction or recurrence relations may have a stronger or more obvious relationship to the original code. other types of analysis of the algorithm are simplified also.
another main advantage is syntactical simplicity. parsers for other languages are very complex but the Lisp parsers are very simple. so Lisp is a great language to study the theory of parsing in.
another key aspect is analyzing software more from a logical or mathematical lens/view rather than a "computer-scientific" perspective.
as the other answer points out, Lisp is all about recursion instead of iteration and recursion is very much at the heart of CS.
more advocation for a "$\lambda$-view" and detail can be found in , a free online and semifamous ref.
 Structure & Interpretation of computer programs, by Abelson & Sussman