TBH sales wise I don't usually touch Jaguar these days, since the hey day when they discontinued the original straight 6 and V12's but I used to do a lot of those back in the 80's and 90's, owned a fair few too.
The V8's I've worked on a few now, there are 3 or 4 running around these parts who are good Customers of my Workshop, normally I'm dealing with Coils, ABS Pumps and water ingress causing Electronic issues on them. My customers have already had Pumps and Timing Chain upgraded Tensioners now.
The V8 Jag's are a money pit, much more so than an equivalent Mercedes
Water Pumps are a common item, and absolutely pour coolant out when they really let go.
You need to find a good indie near you, and don't believe anything a main dealer tells you.
Start with a cooling system pressure test, any good honest mechanic should be able to do that and confirm if it is Pump or not, then go from there
HTH