Good that you've got somewhere under cover to keep it - I didn't have that luxury when I was having all my fun and games and it made things a right chew on!
I would be cautious about feeding anything down the drain pipes, as you'll have seen all the drain tubes are connected to a knuckle which is a friction fit onto the roof tray.
I've seen a couple of horror stories where people have fed Bowden cables down and pulled the drain tube off the knuckle. Similarly using compressed air, read a guy who blew the tube off the knuckle and had to go to the faff of putting everything back together with sealant!
I've used plastic hedge strimmer line, doubled up into a loop with decent effect. It's got enough strength to break up the gunk but it's malleable enough it will just stop if it hits the drainage tubing. I prefer not to put anything down the drainage tubes though, I now tend to just pour some warm water mixed with mild detergent down there every few months.
it may be different in the superb, but on the Octavia the main thing is the rubber nipples on the bulkhead. They simply need squeezing and manipulating (ooo err) a few times and the crud that comes out of them is quite surprising.
By the sounds of it, It may well be that the roof mechanism itself is gummed up on yours.
When I first got mine it made quite a racket opening/closing the roof and it sounded like it was struggling - it simply hadn't been maintained at all and the car was about 6 years old when I bought it.
I blew the the rails and mechanism out using compressed air, then clean them with a soft paint brush before running a shop vac around to try and pick up anything loose and a final wipe down with a microfibre cloth. I then give everything a coating of WD40 Specialist PTFE lubricant. After that the mechanism was running nice and smoothly and much quieter.
I do all the above annually now and touch wood everything runs nice and smooth. I could probably get away with doing it every couple of years to be honest but for the sake of an hour or so's work, it's worthwhile.
It may be worth you trying similar, see if yhr roof mechanism frees up at all with a bit of maintainace.
There's another test you can do at home, even with the roof not closing properly which will help to see if there is an with the actual integrity of the roof itself. Simply pour some water into the plastic overflow tray which surrounds the roof.
With your newly clear drainage tubes, this should drain away without issue. If however you get any water ingress Into the car then you might have a similar issue to me where the water is leaking from the plastic tray (in my case it was running across the roof liner and then down the outside of the drainage tubes).
As much as they're a nice thing on a summers day, I would never have another car with a sunroof of any sort ever again, it just isn't worth the faff and potential bother and expense.