What is the problem with regex when checking at least 1 uppercase letter, 1 lowercase letter, 1 digit where the expression can be in no particular order

The problem is that Regular Expressions are text processors, not logic engines - they aren't designed to do rules based operations.
And that is exactly what you are asking for: to do counting and act on rules.

So by all means use Regexes to identify if characters are present in a string, but process the rules based on that in code, don't try to do it all in a text processor.

Think about it: even if your could do exactly what you wanted, what happens next week when the rules change and require a special character as well from a specific list? Implemented in code it's trivial to understand and modify - which is not the case with any regex beyond the trivial examples!

Regex is a tool in your box - but that doesn't mean it should be used at every opportunity: a hammer can be used on screws, but a screwdriver does a better job!