Though I am sure that @cardinal will also put together an excellent program, let me mention a couple of books that might cover some of the things the OP is asking for.
I recently came across Probability for Statistics and Machine Learning by Anirban DasGupta, which appears to me to cover many of the probabilistic topics asked for. It is fairly mathematical in its style, though it does not seem to be "hard core" measure theoretic. The best "hard core" books are, in my opinion, Real Analysis and Probability by Dudley and Foundations of Modern Probability by Kallenberg.
These two very mathematical books should be accessible given the OPs background in functional analysis and operator algebra $-$ they may even be enjoyable. Neither of them has much to say about applications though.
On the more applied side I will definitely mention Elements of Statistical Learning by Hastie et al., which provides a treatment of many modern topics and applications from statistics and machine learning. Another book that I will recommend is In All Likelihood by Pawitan. It deals with more standard statistical material and applications and is fairly mathematical too.
On the grounds that you want something (a) well-motivated, (b) less dense, and (c) introductory (undergraduate or early graduate level), you might want to consider a text like "Mathematical statistics and its applications" by Larsen and Marx. The "and its applications" is important because the authors give a practical motivation to the theory that you may have found missing in Casella and Berger. This is still a "mathematical statistics" book though, not an applied practitioner's guide on how to apply statistical methods that are otherwise treated as a "black box". There are exercises in Minitab, which I am sure you could translate into another statistical language of your choice.
It only covers a small fraction of what C&B do, and it may not be "pure" enough for your tastes; perhaps you will find the applications a sort of contamination rather than motivation! But C&B is quite a heavy book to hit, if it's the first that you take on. Larsen and Marx is (in my opinion) quite clearly written, covers simpler material, and is very well type-set. That all should make it easier to get through. Perhaps after working through a book pitched at this level, it would be easier to mount a second assault on C&B or similar.
The reviews on amazon are pretty mixed; it's interesting that people who taught courses using the book were generally pretty favorable (one criticism is that it is not as mathematically rigorous as it might have been) while students on courses where the book was a set text were more negative.
If you would prefer a text that was more mathematical in nature, then I think you might need to work on your background knowledge first. I can't see how it is possible to understand a rigorous proof of the Central Limit Theorem without a good background in analysis, for instance. There are some "intermediate" texts, of which Larsen and Marx is one, which are not so rigorous as to be incomprehensible to someone without an analysis background (so you get a "sketch proof" of the CLT rather than a formal one, for example), but which are still "mathematical statistics" rather than "applied statistics". I suspect your basic choice lies between the more mathematical approach, or reaching into statistics via this sort of intermediate-level book. But if you want to take things higher, then at some point you are going to need some more mathematics.
MIT runs a course for introductory statistics for (undergraduate) economics, with a set text of "Probability and Statistics for Engineers and Scientists" by Sheldon Ross, and recommended texts of Larsen and Marx or alternatively DeGroot and Schervish, "Probability and Statistics". The MIT course authors compare them as:
Larsen and Marx's book is a bit more chatty than Ross', while DeGroot and Schervish's is a very good book but somewhat more difficult
If you want something antithetical to the dry style of C&B then the chattier style of L&M might suit you. But those other suggestions for texts of a similar difficulty level might also interest you.
Best Answer
When I first studied probability and statistics many decades ago, a probability course based on William Feller's "An Introduction to Probability Theory and Its Applications, Volume 1" was prerequisite to the statistics course that used Hogg and Craig. With Volume 1's restriction to discrete sample spaces, Feller achieved considerable rigor without needing measure theory. Even in Volume 2, his use of measure theory was generally approachable. Both Volumes are still available, over 40 years since the author's death.