Even twitter and soup.io are threaded, if you look at the metadata. A twitter reply references its parent, and you can find replies to the post (even when it mixes direct replies and later ones, but you can reconstruct the tree by looking at the parent-relation). Soup.io allows following threads by clicking "this is a reply to ..." or "replys to this post". Your example is a bit flawed, because it displays a thread not as a real thread, but flat with nesting. As i said before, look at the heise-forum. You read one post at a time, and the thread-structure is displayed below, so you can decide which post you want to read next, either following the sub-thread or jumping to the next reply of the parent-post.

Heise Forum would never be the same without threads. And its hard to discuss some topic, while others are shooting replies to much older posts in between. For serious discussions over a time longer than a few hours (in a high frequented forum), you just need to use threads. even stackexchange has threads ... each solution has its own comment thread. Of course, its not possible to have there a full discussion system, but you said yourself, you do not want to provide a forum there.