Over in the 500 Classic section there is often mention of modern replacement capacitors going faulty soon after installation or even not working new out of the box. Someone, possibly 'fiat500' in the course of his investigations came across some useful information about modern capacitor construction and how/why they fail so soon and gave a couple of links, worth reading.
If that dealer fitted alarm system is still connected, I'd try disconnecting it, alarms are notorious for going faulty after a few years. Also, check that no other anti-theft devices are fitted.
You mention that a new fuel pump and sender unit were recently fitted - you don't say how long you've had the car or if it has had this problem of cutting out when you bought it.
'jackwhoo's suggestion of wiring up the coil direct to the battery and also fitting a new low tension lead is I think a good idea

- this will remove any possible interference that the alarm/immobiliser might be having, as well as ruling out any issue with the ignition switch (which you've already tried), coil low tension lead or possibly faulty revcounter.
Re:- how to check for a spark if you've no one to help. With the distributor cap removed, engine turned to where the contact breaker points are closed,
car out of gear, coil lead disconnected from dist. cap and held circa 6mm from a good earth on the engine, just flick the points open with a small screwdriver - this should result in a good spark between the end of the coil HT lead and earth. Remembering that this is difficult to do on an X1/9

(iirc there is a removable access plate behind the spare wheel? inside the cabin, might only be on the earlier 1300 model?) instead of trying to open the points using a screwdriver, you could alternatively remove the nut holding the low tension (distributor) lead onto the coil (iirc it's held on by a nut), leave the terminal sitting on the coil threaded stud, switch on the ignition to allow to coil to 'build-up', then after a few seconds, remove the low tension lead from the coil stud (use an insulated pliers)- this will have the same effect as the c/b points being opened, i.e. the magnetic field in the coil will collapse. resulting in a big spark between the coil ht lead and earth. Doing the check this way also has the advantage of eliminating a capacitor shorting out when hot from the equation. The main purpose of an ignition capacitor is to reduce wear on the c/b points due to arcing, it also has another purpose which evades me at the moment...
I don't think I'd bother either with checking the valve clearances just yet, they rarely closed up much on the sohc or dohc Fiat engines. If they were closed up significantly, I'd expect performance to gradually drop off, be difficult to re-start when hot, maybe to backfire through the carb if an inlet valve were affected etc., not for the engine to cut-out relatively suddenly (I noted the c. 10 sec delay you mentioned). You could perform a cylinder compression check both cold and hot, and then again after the engine has cut out but iirc, it's not recommended to remove spark plugs from a very hot engine... There is a way of using a vacuum gauge to spot tight valve clearances but I can't remember what readings you would expect if this was happening. (might be a momentary dip in the reading if only one valve is affected, I don't know if it would indicate if all valves clearances were tight).
You mention that your carb seems to get quite hot (I'm thinking fuel vaporisation as a consequence) and that the engine bay temperature is very hot, what is the engine coolant temperature? If this is also high it might be indicative of tight valve clearances, particularly if the coolant temperature increases just before the engine cuts-out. If I thought a carb was running to hot, I'd try cooling it down by spraying it with cold air e.g. by using an airline/ leaf blower/ shop vacuum set to blow air, or spray it with water. I suppose you could bring a spray bottle of water with you and use this to try to cool down the carb and fuel pump and see if the engine then restarts.
One frequently used way of tapping into the inlet manifold vacuum is to tap into the vacuum pipe going to the brake servo - but the X1/9 doesn't have a brake servo? Another way used to be to tap into the vacuum pipe going to the distributor vacuum advance unit?, but the X1/9 doesn't have a vacuum advance unit on the distributor either, well not on the U.K spec models anyway. Yet another way used to be to fit an aftermarket carb spacer (insulation) plate that came ready-fitted with a hose stub - these were available for a variety of carbs - I don't know where you get one nowadays or if you'd be bothered trying to get one/make one? You did mention trying to get and fit an additional heat insulator?, so maybe modify your existing insulator plate to accept a pipe stub to connect your vacuum gauge onto?Probably easier to carefully examine the inlet manifold to see if there are any blanked off connections or existing hose connections that you could tap-into?
Don't be fooled into assuming that fuel in the fuel filter equates with sufficient fuel in the carb float chamber, it might not be if vaporisation has occurred, pumps can't suck vapour, they're not great at pumping vapour either, so any fuel in the filter won't be pushed into the carb. Ditto, the fact that you get fuel spraying from the pump jet doesn't mean there is sufficient fuel in the carb. float chamber, the accelerator pump has it's own little store of fuel under it's operating diaphragm, try opening the throttle a few times to check if the accelerator pump can keep delivering fuel, i.e. that there is sufficient fuel in the carb. float chamber to replenish the acc. pump circuit. There might not be, if fuel vaporisation has occurred, although the engine should attempt to start after you squirted fuel in by operating the accelerator pump jet.
I still don't know about the wisdom of using these coils that have 20kOhm secondary resistance - I always associate these as being for use with electronic ignition systems, can't you find a replacement coil that is close to the original Fiat one or as 'jackwhoo' suggests, just refit the original (it hasn't been proven to be faulty and replacing it with a 20kOhm sec. one has helped).
That 'cyclonic valve' looks to me to be a simple engine breather housing - there's nothing inside except a central tube, it's purpose is to separate oil from any piston blowby, so that only 'fumes' enter the breather pipe connected to the airfilter. It practically never gives any trouble, doesn't even need occasional cleaning out - all you'll find if you remove it is a little harmless varnish coating assuming the engine has had reasonably regular oil changes. I don't have a Haynes manual for your car so I can't check out the control valve on page 62, at item 4. If no drive belt is fitted for this device, then you probably don't have one. The X1/9 was marketed in various countries, some of it's specifications/equipment were varied to suit, you should see what one destined for use in 'dusty conditions' had fitted, I worked on one that should have gone to Australia...:bang: Just be glad that you don't have some of the rubbish fitted that's mentioned in the manuals, less to go wrong.
Can't you get the cut-out problem to materialize on roads other than motorways, dual-carriageways etc? Would sure making carrying out some checks as soon as the engine cuts-out a lot easier and safer.
The only other advice I can offer is to leave the carb. alone until you've positively ruled out an ignition problem.
I find it strange, that the only improvement came when you hot-wired the ignition, you said the car restarted at the 2nd attempt and didn't cut-out again. Why not just run the car in this mode for a while, try driving for more than 45 mins and see what happens.
Sorry for the lengthy post, some people have suggested I should write a book, but I can't seem to find anything to write about...
Al.