I could almost care less if an E-type actually ran. Mechanical is child's play compared to the potential tub pitfalls. You will want the car up on axle stands to do the inspection.
Unless you can be sure that the paint is original, you need to divine what is under the paint. The sill structures are everything on a jag. A borescope would not be going to far when doing this inspection. The good news is that body-men are lazy. Shoddy body-men even more so. Look for the transition areas where one piece of the tub meets another; places where it would be time consuming and physically uncomfortable to do the detail work if you were doing a filler-job on the car. Low on the sills as they disappear under the car is a good area. Same at the rear valence. You are looking for areas where filler has not been fully sanded to blend in with the metal.
Even if you don't find the obvious, be aware there are some Bondo artist and don't let your guard down.
Assuming it has had some rust repair, some of the welds must be visible. Seek them out as a clue to how well the work was done. Are they bird poop and ground or half decent puddle welds? If the weld seam has been plastered in seam sealer, then you might want to know why?
How does the bonnet fit? Not worth continuing your existence if the fit is off and you have to sort that. I'm sure Hemlock potions was a British Leyland accessory for dealing with this.
Warm it up and run the heater. If you see any drips in the cabin coming form up around the firewall, be aware that pretty much the whole car has to come apart to replace a tube that runs across the firewall. Back in the day before they became icons, people used to scrap the cars for just this reason once they got the quote to repair. Not sure how you test if the car wont run: Pressure test maybe?
Look for where the torsion bars tie into the chassis, this needs to be in perfect shape.
One that people messed up when the car was new was mounting stuff to the engine truss. The metal is very thin and the act of drilling any holes in them compromised the delicate engineering that went into them. So go armed with some pictures and look for extra holes.
Interiors are expensive to restore, so be critical.
Mechanical I assume you can asses on your own. A leak-down and compression test is the tell-all for me. They are fairly tough engines and the later boxes are not bad. The carbs are British, expecting them to work is delusional but not really a big deal to sort other than a nuisance factor. Exhaust get a bit spendy. Like all Brits they overheat if the system is no in perfect condition.
If you are just looking for the restoration challenge of a lifetime, your own personal Mount Everest to climb, then ignore all of the above and just buy whatever is there for as cheap as you can get it. Sort it on the fly.
All of the car is built like that. I think they hail from a time when excess complication was seen as a sign of "quality"
