Weakest link - a different question?

Al trusts his technical instinct, I trust my 'audiophile' instinct.
First, wall current is 50% of success unless power supplies of the electronics are so robust and sophisticated that that makes virtually no difference. I am not mentioning batteries. This probably is never the case. So that's the first thing to reasonably get under control. Your Panamax is unlikely to cut it.
After that - the source. It is a driving force, if you don't feel it - no good. After that - speakers. To hell with the room to begin with - listen near field. You don't like their character - throw them out. The power amp or integrated. How well does it deliver that driving force coming from the source ? Then wires. Phono stage is part of the source.
It's not all that difficult with some experience, good ears and opportunity to try out a few components. If the latter is not possible it of course is more difficult.
My point is - there is no methodology in a scientific sense of it. It is too complex for any device to figure out but brain. Any computer is infinitely inferior to brain and mind. 

BTW, the best answer is: They ALL are to some degree deficient and in need of upgrade. Change ANY of them, for in nearly all (like 99%) of audio systems they are not optimized and there is room for improvement. People are guessing at what's missing, what's wrong, what needs to be done. Then, and here's the best part, they change it and what do you know? They were confirmed in their assessment because it got SO much better. Little do they know that because of the relatively lower performance of the system ANY component or cable or speaker could confer a sizable improvement.  

The exception to this is when you get to such a high level that you really are knocking on SOTA quality, or conversely when your budget dictates how much you spend and you have tapped out the upper echelon of gear in that price range. But, you have to go through a lot of gear to make a solid assessment that you are in fact at that point. Most people never get close to that. 

So... change whatever you like. It doesn't matter. You will, if you do your comparisons or research properly, get a nice bump in performance, and you'll feel like a genius.  :) 

Without another component you have no basis for judgment. You need to do comparisons to find a relative performance threshold for a piece of gear. Apart from that you have no clue how each piece is performing. 

Didn't Elizabeth start a thread very similar to this question a couple of months ago?

It is an excellent question and sometimes difficult to diagnose.  After many years in this hobby, I believe strongly that the speaker/room interface is the most overlooked item in creating a system.  IMO, in a normal domestic environment where speaker placement is limited, there is absolutely no way to tell how a speaker will work in your room without a home demo. I have had many very good speakers in my room including Totem Forest and Hawks, Harbeth Monitor 30 and SHL5s, Coincident Partial Eclipse, Opera Callas Monitors and now the Trenner & Friedl Pharoahs.  Each one worked to a point, but when I put the Operas up it was like magic.  The speaker worked perfectly in the room and those small speakers filled the room like a concert hall.  So the problem is - how do you do that?  IMO, either get a dealer to allow you to demo, or buy used, hope for the best and sell if they don't work out.  But once you get the speaker/room thing together, everything sounds a lot better. 

@almarg thanks, your first two points make eminent sense and I hadn't identified those strategies so clearly. The rest, yes, I'd reached similar conclusions.

An excellent question. In my case I combine a multitude of approaches:

1) I compare results listening via speakers with listening via my very revealing Stax electrostatic headphones. Headphone listening, of course, takes the room, the speakers, and the power amp out of the picture.

2) I compare CD and LP playback, using the same recordings to the extent possible.

3) I do **a lot** of research and reading.

4) I only consider upgrading when a broad sampling of what I consider to be credible reports about a particular component causes me to become particularly excited, and suggests to me that the upgrade is likely to provide an improvement that is more than marginal.

5) I try to avoid components for which either user reports or technical considerations lead me to believe that the component is particularly critical with respect to setup or system-matching. Increased criticality of setup or system-matching can only increase the difficulty of identifying a weak link.

Also, for that matter, I avoid components that seem to inspire controversy with respect to either performance or the manufacturer’s customer service. Alternatives are usually available which can be counted on to not entail such risks.

6) I trust my technical instincts, as an experienced designer of sophisticated analog and digital circuits (for defense electronics, not for audio).

Regards,
-- Al