If you search PPRuNe for "Edinburgh ACN PCN" you will find a number of posts in this thread, and in a thread in the Tech Log forum, that discuss the strengths and weaknesses of EDI vis-à-vis regular long-range wide-bodied operations.
A brief summary is;
Runway 06/24 is long enough for a decent selection of long range wide-bodied routes with a decent selection of wide-bodied aircraft types but the technical data published during the BAA era indicated that it was neither long enough nor strong enough for all such aircraft to operate at their respective max take-off weights. That said, the most likely potential medium / long-range routes from EDI wouldn't require take-offs at maximum weights or require a longer runway. By way of example, during the BAA era, a reasonable selection of medium and long-range routes would have been possible with non-winglet B763s (using existing main apron stands) and with A332s (if a suitable main apron stand had been created for this aircraft type). A stand for larger wide-bodied aircraft might have been possible by, for example, reinstating stand 6A and/or reinstating the diagonal stand that used to exist between stands 11 and 14. Other potential locations for stands for larger wide-bodied aircraft are conceivable
One of the significant limitations I established after reviewing detailed technical data published by NATS, Boeing and Airbus was that the PCNs of the aprons and (to a marginally lesser extent) the runway (06/24) that were published during the BAA era limited the payload / range of a B773ER quite significantly unless ops were sanctioned with a significant ACN>PCN overload. I checked with BAA to see whether my conclusions were correct and they agreed with my assessment. These PCNs would presumably have been one of the factors that influenced Emirates decision to operate from GLA rather than EDI but other considerations might have also been important. I’m not party to Emirates deliberations on this subject but the PCNs and other facilities at EDI for such flights at the time that Emirates made their decision were demonstrably inferior from an operational perspective (including potential payload / range) to those available at GLA at the time.
As has been pointed out, the SE apron has a number of stands capable of handling larger wide-bodied aircraft however the PCNs of taxiways Lima and Mike that link that apron with Taxiway Alpha had a published PCN, during the BAA era, that prohibited regular use of these stands for long-range wide-bodied ops. B773ERs to Paris and A332s to Dublin for rugby charters and certain other wide-bodied flights were permitted as one-off ops on an occasional basis but take-off weights (and therefore ACNs) for most of these flights would be low compared to long-range ops.
Three main apron stands were (and still are) able to accommodate aircraft up to TriStar / A310 / B752W / B763 (without winglets) size. One of these was almost exclusively a BA shuttle stand. There had been various stands on the main apron over the years that had been able to accommodate aircraft up to B744 size but terminal alterations and apron re-layouts by BAA eliminated all of these at various times over the past 40 years. Perhaps one or more of these stands could have been retained / reinstated (as mentioned above) but BAA chose not to.
Delta operated regular B763 flights from EDI for a time but the arrival time in Atlanta was quite late (IIRC it left EDI at 13:45). Every time I considered using this service, my onward connection arrived in my ultimate destination in USA just before or after midnight. By the time I would have collected my bags and reached my hotel it would be about 2 a.m. or later. As a result, I never used the service and I suspect that many other potential passengers felt the same as me. Whether the arrival / departure time at EDI was constrained by the availability of EDI stands/ slots or ATL stands / slots or some other reason I couldn’t say for certain, but the two stands that were able to accommodate the Delta B763 were heavily utilised during the morning for B752 ops and, too often, the Delta B763 would arrive and no suitable stand would be available. My suspicion is therefore that the Delta B763 arrival and departure time was constrained by a shortage of suitable stands at EDI.
It's interesting to note that during the week that GIP took over operations at EDI, the PCNs of many parts of the airfield were significantly upgraded without (as far as I know) any physical works having been undertaken. In the period between then and now, further PCN upgrades have been declared (via NATS) and I expect that more will follow quite soon.
Whether or not BAA knowingly understated the PCNs during the preceding years I couldn't say but I was somewhat surprised to see circa 45 year old pavements being upgraded from C strength sub-grade to A strength as a result of a change of ownership before GIP had carried out any physical works. One possible explanation is that GIP technically re-assessed the PCNs of these sections of pavement (perhaps during due-diligence surveys prior to the purchase) but perhaps there is another explanation. It is not clear to me why BAA had not carried out this technical re-assessment many years sooner given their repeated press releases about trying to attract further long-haul flights.
Regardless of whatever BAA did or didn’t do during their period of ownership, the good news for passengers that use EDI is that GIP have invested and continue to invest in PCN and other upgrades and are planning to invest in the stands necessary to allow regular long-range wide-bodied ops. I hope this investment provides them with a suitable return on investment and that EDI’s long-haul route network will see further expansion in the not too distant future.