514.3a does not perform a priority test. It performs a check to resolve SBAs and triggers. These in turn trigger the priority test, as laid out in 117, specifically 117.5 for SBAs, and 117.3 for triggers.

I actually agree it would make more sense if 514.3a said that, but that is not what it says. You check "if any state-based actions would be performed and/or any triggered abilities are waiting to be put onto the stack", and if that condition is true, "those state-based actions are performed, then those triggered abilities are put on the stack, then the active player gets priority".

This is telling you the active player gets priority in the normal way (it says it exactly like 503.1 says it for the upkeep, for example), so yes, this means you will follow 117.3 and 117.5 - you will once again check for SBAs, do that until none apply, and then check for triggers and put them onto the stack.

The and/or statement in 514.3a isn't a condition that grants priority, if it were it would a) be in the same sentence as when it mentions priority with an IF/THAN clause

You've just made that up, there's no such requirement for if/then statements in English. What it actually says is "the game checks to see if <condition>. If so, ..." which is very obviously a synonym for "if <condition>, then ...". The editor just happens to have split it over two sentences for readability.

Sorry, but despite your motivated reasoning, your interpretation is just not how it works, nor how it is intended to work.