I've used this Schwalbe G-One Speed tire for a long while with no problems. I got 2 punctures in the last 2 weeks though. First was because of a 2 mm thorn, second one was of unidentified cause. While looking carefully I noticed that tread is worn out but not to the point of being unusable. I switched to another tire meanwhile but I'm wondering as a general rule, do people notice an increased frequency of punctures with tread wear?
Yes, the less rubber there is between the tube and the road, the more likely it gets for you to find a shard/stone/thorn that's big enough to cut through to the tube. The amount of rubber is your main defense against punctures.
Consequently, puncture proof tires usually add quite thick layers of extra rubber / rubbery material. They may also have improved protector layers, but the extra rubber is their main defense.
And the effect of these extra rubber layers is huge: When did you last have a blow-out on your car (if you have one)? Cars do much more kilometers than bikes, they are driven across the same kinds of surfaces, they have a much wider track, allowing them to collect much more shards/stones/thorns, but they virtually never have punctures. Simply because their tires have such a massive rubber layer.
So, yes, the more you wear down your tire, the more frequent the punctures will become. If punctures get too frequent, think about replacing the tire. Ideally with a serious puncture proof one.