Tread isn’t such an issue
I’d back Dave, all the Schwalbe tyres I’ve ridden have been good – the Stelvios are durable and grippy, and you can pick them up for about £15 a pop, Aside from Dave I’ve heard good things about the Ultremos, including someone who had a bit of a ‘moment’ on an alpine bend a few weeks back and reckoned they played a big part in getting him through in one piece. The other tyres, I’ve ridden recently that have impressed are the Michelin Pro Race, you can pick them up for just over the £20 mark which puts them at the same price point as your Contis and the Schwalbe Ultremos.
One thing I would say is that I wouldn’t get too hung up on tread, on road tyres it’s pretty much an irrelevance in terms of grip, compound, tyre pressure, and the size of the contact patch are what count. The contact patch on a road tyre is so small that there isn’t enough of it for tread to make any difference, but the small amount of tread there might be just takes rubber away from the contact patch – which probably isn’t ideal.