So I have been on here with lots of threads, research, opinions, etc....I have the solution now to an intermittent overheating problem I have suffered for about 3 months now, car has been parked for a good amount of that time....
xk8 convertible with 46,000 original miles, i am owner #2, car sat a while in a garage and was owned by my cousin who twisted my arm over several years to buy it from him, lots of deferred maintenance issues. it would not pass smog either.

car is well sorted now, well as a 20 year old jag can be. everything works but the trunk lights

car at that time began to over heat within seconds of being just fine....i had already replaced the water pump, thermostat , coolant pipe and sensor 6,000 miles ago myself....i did the blue stuff with the thing you squeeze test for head gasket, no change in color, the car was over heating and not really over heating, no message in the display....i was baffled and got all types of thought from here and other forums i asked....no other symptoms like water in oil or oil in water or white smoke....i gave up and parked it and gave it allot of thought....

Found a local shop, Bosch on Spring Street, here in redwood city, ca. Matt knows these cars and other European cars as well. you can eat off the floor it's so clean and there are many different marques of cars there. i found him by hit or miss...

Took car in with the elusive issue...and found the cause and the fix....here we go....
the whole project took 5 days, period

my thoughts on cause....
about 3 months ago the coolant sensor came loose from the bottom of the expansion tank....at this point i replaced it, don't loose the clip for the wire or you are in trouble, found the solution myself with the help of someone on the forum, thank you. parked the car downhill one day and saw a small puddle under car, but no apparent leak...i was loosing some coolant but now allot or always and it would over heat and cool off and over heat...and the fans always ran just fine...WTF....

finally i gave up, put my ego aside and asked for help...first they could not find and thought lets throw parts at it, i said no, do a pressure test of system, they let car sit...sure enough there was a small puddle next day,a bout 1/2 cup full....examined...found water pump leaking....ONLY under load and after warming up......

i ok'd a water pump even though i put a cheap chinese made amazon one it is 6,000 miles ago...it leaked and the gasket became toast...so when the car leaked only after warm up, and under pressure it over heated, intermittently....first symptom was expansion tank blowing out the sensor underneath which is pressed into the tank....

i ok'd a water pump, new coolant, thermostat, few other minor things...$300 labor for diagnosis and fix, and another similar amount for parts, tax, etc.


car is back runs like a top and i am soooo happy, i may even keep it! wow what an ordeal. my skills as a self taught mechanic are waning in my golden years and my ego refused to allow me to face it was parts i had put in recently that went bad! stay away from CHEAP COOLANT parts also. hope this helps someone with a similar problem

 

I am happy to read the above, but I frankly don't see the relation between a slightly seeping waterpump, and the overheating issue, unless the cheap "Chinese" pump was broken upon installation.

 

so air gets into the system when there is any leak in the cooling system, this air eventually displaces coolant and causes air bubles (as in burping the system), coolant does not flow or reach upper reaches of engine at first, this causes over heating of upper portion of engine which causes the car to think it is completely overheating, and it's all bad cause the gauge just goes ballistic...

so car over heats, but not when cold or before the thermostat opens which causes over heating intermittently after warm up ...sometimes going back down to normal after new coolant i pumped beyond a bubble for a short time....

so air got into the system when engine metal expanded near water pump causing small leak, only under load and only after warming up and sitting for a while.

 

Not intending to dispute your thoughts, or an "all know better", just a bit of 'devil's advocate' here for the sake of discussion and saving your engine ....
I am also not 100% sure on the exact XK system, but your's being a 1999, it is fairly similar to my X308 I presume.

The coolant system is pressurized due to the temperature which rises if the engine runs.
A bit of that pressure is absorbed by the system, then more is absorbed by the engine bay expansion tank, which then releases any overfill in a secondary (atmospheric) tank.
If your engine cools down, the reducing pressure sucks the secondary tank back empty, basically providing the "always full" situation.

In your case, the pressure pushed out some coolant through an otherwise tiny hole.
If any air was later sucked back through this hole, it was meaningless, specially as the coolant system is designed to auto vent any bubbles.
Worst case scenario is that the air allowed through this hole balanced the vacuum created during cooling down, hence not sucking sufficient coolant back from the atmospheric tank.
That should result in a drop in the level at the engine bay expansion tank for every heat cycle.

If it was me, I should still suspect there is a problem, unless it was caused by the very bad functioning waterpump itself.
Sorry for the bad news, but you better stay alert a bit longer.

BTW, I myself run the Torque Pro app most of the time on the Jaguars, and I set the coolant alarm at 100C/210F (flashing dial) and 105C/220F (submarine frightening horn).
The gauge in your cluster is absolute worthless, and will only jump up minutes from disaster.