does it teach analysis of algorithms or its foundations in a manner that is useful and generally applicable?
You may be interested in the companion book The Concrete Tetrahedron
The book treats four mathematical concepts which play a fundamental role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates.
Their key features, in isolation or in combination, their mastery by paper and pencil or by computer programs, and their applications to problems in pure mathematics or to "real world problems" (e.g. the analysis of algorithms) are studied. The book is intended as an algorithmic supplement to the bestselling "Concrete Mathematics" by Graham, Knuth and Patashnik.
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You are probably looking for Sipser's Introduction to the Theory of Computation.