Stand Wave Ratio is a bit of EM waves basic theory, esoteric as it to most people. Basically what we are measuring is the magnitude of the incident wave with respect to the reflected wave. A 'standing wave' can be either or both a voltage or current depending on what you're looking at. Essentially, though, it's giving you a measure of the impedance match of the radio to the antenna system.
The output circuit in your radio has a tuned frequency response, so the expected load is not necessarily resistive. The problem is impedance is not an exact fixed value like a resistor but is reactive in nature, so it's got a capacitive, inductive and resistive component and is frequency dependent. That 50 ohm output is based on a relatively narrow range of frequencies. The antenna has an impedance, too. Ideally the antenna impedance should be exactly 50 ohms at the frequency you're transmitting and receiving on. There is only one frequency where the impedance is exactly 50 ohms and this is where bandwidth comes in. If an antenna has a narrow bandwidth, the impedance changes rapidly as you deviate from your center frequency and on the opposite hand a wide bandwidth antenna has a broader region closer to 50 ohms. The radio does not want to see any of the power being reflected back for various reasons, but in the end it's so you don't damage anything. BTW, the load can be 100% resistive and the radio won't complain. This is called a dummy load and it converts 100% of the RF energy into heat with no EM waves being transmitted. So in a real antenna you want a minimum of resistance, which is just lost to heat (which is one reason why a true 1:1 SWR is never achievable, you lose some energy to heating the copper of the coax and the steel of the whip).
Relative to the radio, the feedpoint of your antenna is the connector at the radio end of the coax. So you are 100% correct in saying that the coax needs to be accounted for. But the thing is that the coax is primarily just a transporter of energy and should be as close to invisible as possible as long as the end connections are solid and the coax does not experience kinks, cuts or deformation. If your cable is physically intact, it will have a very, very low reflection coefficient and in real life it's effect is actually pretty small when compared to the antenna itself. Modern antennas have a fairly wide bandwidth and are quite tolerant tuning wise. Not to mention that SWR is not a linear measurement. So at 1:1 (unachievable in real life) is 100% energy transfer, 1.5:1 is 96% power transfered (i.e 4% power loss), 2:1 is 89% power transfer, 2.5:1 is about 80% power transfer and 3:1 is 75% power transfer. So the range of ratios that give you acceptable performance is generally 2:1 and closer, which means you have a pretty big tolerance when you read the meter.
The problem is that when they say they are factory tuned, that does assume certain things, like coax type, good ground planes and such. Each installation is unique and so there is nothing that is a truly perfect 'tune'. They can assume a few things and get it close, which when coupled to a wide bandwidth will mean that the antenna will not exhibit a high SWR in most installations. The bottom line is this, though. I've tried a few different mounting locations and found that neither my stubby Larsen 2/70SH or the Comet C767 needed much tuning with different coax lengths or mounting locations. So as to the question of needing an SWR meter, I dunno. I have a couple of them and feel better about checking the SWR at low power after getting done. It's a sanity check to verify the connector is right, the mount is grounded right, the coax is not kinked, etc.
Edit to add: I might clarify and say that I could get a couple of percent better power transfer by tuning. My Comet for example only gained 0.1 on the SWR when I tuned it. The Larsen 2/70SH (this is a short antenna with a spring, really short in fact) didn't change at all but it's only about 12" tall and so the whip doesn't have much adjustment anyway. It has terrible range and is only useful for trails and in the city so I don't break my antenna.
