You're right, the runway is at 622'. However, the default P3DV3.3.5 ATC (as with all the previous and MS FSX etc) when flying IFR always brings you down to 2,500 for approach, seemingly because that is the height at which the ILS will lock on. Anything higher and the ILS doesn't work.
See below (changing FAF alt), but an alternative way could be to increase the FAF distance. This can only be done globally in Options #2 -> Advanced Options -> FAF Distance. I have set it to 9 (from the default 7) to avoid similar situations without changing the FAF Alt for individual airports. Please do not change the default FAF Alt here unless you know what you are doing, but see below to adjust it for individual airports:
Thanks for the tip about configuring the airport altitudes. Now I have to figure (pun intended) out how to do it!
On the SIDs/STARs page, enter EGGD in the airport field, press Enter.
Now use the up/down arrows next to the FAF alt, and finally Save.
You didn't address my query about the frequent heading changes. Some of which, it seemed to me, the controllers were actually searching for the airport.
This sometimes happens to me if the runway changes, or if I am too close to the airport and they want to bring me back to the FAF.
It's strange that I have no difficulty anywhere else with the lining up. Just my luck that I use EGGD as my home airport. I suppose I could change to Exeter.
With above adjustments, EGGD should work fine, except for the 26/27 confusion.