There may be other factors causing the engine temp to increase when idling (for prolonged periods of time). If the temperature goes back down to a more normal (less hot) level when driving, but creeps up when driving at very slow speeds (as in stop and go congestion), or prolonged idling, you may well have an air flow problem over the radiator cooling fins. That can be caused by a few things, The first thing I would do is make sure none of the cooling tube fins are bent over, thus blocking off the ability of outside air to freely flow through the cooling fins. Also, look for debris blocking the cooling fins (leaves, dead insect carcasses, mud, etc.).
This is something I feel is well worth sharing... On our 73 Mach 1 someone replaced the original standard radiator used with its factory 302 engine with a much wider radiator meant for a larger engine (351/429?). But, they did not change the fan shroud. They just left the older shroud in place, which meant a lot of the cooler air was not being drawn through all of the radiator's cooling fins and tubes. That resulted in the engine coolant temp spiking at 225 - 240 degrees until I got the vehicle moving at 25MPH or faster, where the air outside of the too small fan shroud was able to flow through cooling fins/tube surfaces, and the temp would quickly drop to 195 degrees or so. I suspect the prior owner was unaware of the temp issue as he had the original instrument panel with idiot lights. We opted to install a Dakota VHX analog gauge instrument panel, so I was very well aware of the temperature fluctuations and excess heat at idle issue. No doubt he drove like that for years unaware of the jeopardy he was putting his engine into. Not that the engine can't handle 235 - 240 degrees, but why cause that much thermal stress needlessly? Things are better at 195 - 205 or so.
I am certain I could have resolved the excess temp at idle issue by getting a correctly fitting fan shroud. But, no, I just
had to have some fun. I ended up installing a large three row aluminum radiator with a matching metal fan shroud and dual electric fans. I used a variable thermostat to trigger a fan relay to turn on when the coolant temp hit 200 degrees, and turn off when the coolant temp dropped under 200 degrees. It worked beautifully, although it was a fairly extreme "fix" for what was likely really needed had i wanted to just fix the real problem for much less $$$. But, for me any excuse to go to dual electric fans was fine - I have always wanted to do that with a high performance vehicle, just because... It worked out so well that I did the same thing for our 73 Mustang Convertible as well, despite the fact it has no cooling temp issues. Not that I do not feel it was a fine solution, but in your case simply making sure you have a good fan shroud that fits correctly, and checking the cooling fins and tubes for air blockage, may be enough to do the trick. The previously mentioned thermostat and restrictor plate suggestions are also important. And speaking of thermostats I (also) prefer to run with 192 - 195 degree thermostats.
If you are still having heating issues be sure your ignition timing is set to 6 degrees BTDC. Be careful though, the vibration dampeners where the timing marks are located, for cars as old as our 71-73 Mustangs are, will at times begin to slip, and you will find their timing marks to not be accurate. I have my own way (old school) of setting the optimum timing on those old engines, as opposed to going only by the timing marks. But, first you need to do a vacuum test to make certain the distributor vacuum advance diaphragm is not leaking or ruptured (that can cause excess temps also). And also test the vacuum line(s) feeding the vacuum advance diaphragm to make certain it is getting ported vacuum when the carburetor throttle blades are opened. As if that is not already enough to check with ignition timing, pop the distributor cap and make certain the mechanical timing advance springs are working, and see if you can feel them work by rotating the rotor left and right. You should feel the "springy" sensation s you rotate the rotor left and right. Not huge, just make sure the mechanism is working freely. If the mechanical vacuum advance system is sticking or binding it will also cause the engine to run hot.
But, wait! There's more! Oh, yippy skippy... On your thermostat cover you should have a Thermal Vacuum Switch (TVS, aka Ported Vacuum Switch) with three vacuum hose connections. It is designed to let ported vacuum from the carburetor flow to the vacuum advance diaphragm when the engine is running at a normal temperature. When the coolant temp gets "too hot" (I think 225 to 235 or so) the TVS will block off the ported vacuum switch and send intake manifold vacuum (to include when idling) to the vacuum advance diaphragm in an effort to help cool the engine down. You really need to test the TVS to make certain it is sending ported vacuum to the vacuum advance diaphragm when running at normal engine speed when off idle, as failing that you can run into problems. Also, make certain the vacuum hoses are installed correctly. I have attached a PDF file for 1973 that shows how your engine vacuum hose routing should look, for all engine and transmission configurations - beginning on PDF document page 18. The various calibrations for the various 351C engines (2v & 4v) and transmissions begins on PDF document page 66 for 351C engines in Mustangs. Be sure to select the correct calibration from the chart on page 18, unless the calibration number on the valve cover is different due to an engine change. Around page 64 or so the document page #s get fouled as I inserted the two 351C 4v calibrations from a different source.
As far as testing it to make certain the TVS begins to send intake manifold vacuum to the vacuum advance diaphragm, you could do that once your temp gets really high, but I personally prefer to not do a lot with the engine when it gets that hot. In fact, I try to not let thing get that hot if it is avoidable.
I know I threw a lot at you, and I am hoping the ignition timing and vacuum related suggestions are not needed because the cooling fins and fan shroud suggestions take care of the issues you have. But, if you are interested in seeing more info I have some YouTube videos that you may find useful re: the vacuum advance and TVS vacuum testing matters.
I have not done a video on my old school dynamic, performance oriented ignition timing process, and feel it is best for me to do that as a video as opposed to trying to describe it in written words.
The videos I have you may find useful are provided below.
TVS, Vacuum Advance, and vacuum system related videos:
https://youtu.be/0yRh_m7TvxE
https://youtu.be/DtBcK7Lxq1Q
https://youtu.be/sLwXlsHBd4k
https://youtu.be/ohw1oRlcC4c
Vacuum system related, but more the the auto transmission modulator and EGR valve. Still, it may have some worthwhile info for you:
https://youtu.be/6sTLsBJUgCI
Installing a dual electric fan cooling system (parts list is in the video description):
https://youtu.be/T5DrC2zrpIA
Installing an LED light array to show when electricity is sent to the electric cooling fans:
https://youtu.be/ErDOMJ1hOnA
