Ds' secondary school (Shawlands Academy so wrong side of the country for you
) was very welcoming of visits: dh and I were given a tour when ds was in P6 as we were considering a placing request (and they then also facilitated a visit for ds and a few of his class mates who were also putting in placing requests).
It was also very helpful to a MNer who was thinking of coming back to Scotland from India and her dd joining S4: saw them both when they were over here on a fact finding trip. But maybe it's more used to it because it is so international. In the event, they decided not to come back because of the cost of international Uni fees 
I would delve deeper into "exam results" league tables: if you do want to use them, find the ones that include "virtual comparators", ie is the school performing better than would be expected given its demographic.
I couldn't have been happier with Shawlands Academy (ds has just finished 1st year at Aberdeen Uni) despite it not getting a massively high number of Higher passes as those that are capable do well and go onto good Unis to do a range of subjects (including medicine and law). A very high proportion stay on to S6 and do AHs. And more importantly, they encourage sports, arts and other aspects to turn out well rounded individuals who can cope with life after school/at Uni 
So my bit of advice is persevere and try to see the schools to get your own feel. League tables don't give the full picture.
Try emailing the headteacher and/or if there is a depute in charge of "transition" (ie the move between primary and secondary) and see if they get back to you. Explain that you're down South but looking to move North and want to work out where to live.
However, the problem may be specific to Edinburgh where placing requests are (I believe) rarely accepted - hence them being "stricter" on offering visits.