The VNA only measure at 50 ohms. I suppose that the input and output of your device under test is 50 ohms, a lot of times it is not.
If you had a good directional coupler that is at least -20 db and is good at the required freq, you can make an analyzer. You can get the return loss from the directional coupler. But you need good open short and 50 ohm loads to calibrate it. It will give you the reflection. You then need something to measure the reflection, the SA will do this. Then you need to do some calculations to give you the results in dB. The -20 dB refers to the intrinsic capability to separate the forward going signal from the reflected signal. Anything more than -20 dB (-10 for example) is considered bad.
The result is that you get a fancy, pretty darn good, VSWR Bridge, a SNA. The HP 8920 8935 series service monitors can do this and did come with optional connectors and VSWR bridge. Some came with the software installed to make the setup easier. If you are looking for useful test equipment these things are terrific.
The phase of any measurements is more complicated and would depend even on the lengths of the connections. I do not know how to do that.
VNAs have multiple amplifiers in the box including one that usually monitors the output signal strength to compare it with the reflective value, making things more accurate, of course these things have to be calibrated with OSL also. They also have phase detectors. I think there are at least two RF bridges involved, and multiple amplifiers, can't really remember, I did fix one once.
The 50 ohm load should give you a very low reflected signal, say -40 dB. The open and Short standards should give you near 100 % reflection or 0 dB.
The big problem I see is attempting to calibrate or "zero out" the deficiencies in the components. Including the connection cables and adaptors. If the freq range was small, it would be easily manageable. A VNA or SNA will do this for you by pushing a button.
For those who like to check things quickly: A 75 ohm load will give you about 12 dB reflection or a 1.5 to 1 VSWR as opposed to a 50 ohm load that should give you zero reflection and a 1:1 VSWR
I have actually done what you are describing many times just for fun, and it works pretty well at HF, Gets more expensive the higher in freq that you go. However I did pick up a GHz Narda directional coupler at a hamfest for $5. Some of these things are rated in a small freq band but work OK at surprisingly wide freq outside of their rating. I also picked up some brand new small directional couplers that are good to at least 3 GHZ (my limitation of equipment) for a buck each. I can measure the reflection to a designated load between few MHz and 3 GHz using a SA with Freq Gen or even a separate freq gen and SA.
So if I can do it with only two years of engineering school, so can you.
On the other hand I sold two very nice HP VNAs recently for less than 1K each.
While you are saving the money on a VNA, remember the OSL and connectors and proper cables can be expensive and necessary either way, but I think you already know this.
This whole thing leads to membership in Test Equipment Anonymous