If I may add another possible benefit of what "ignore a topic" may offer, sometimes we want to leave a topic we're previously engaged on, either because the topic no longer interests us or it has turned into a meaningless discussion filled with spam posts. I am not sure there is currently a way to "remove" these threads from our "show new replies" page. The ignore a topic button would be a nice thing if it means it will remove those threads from my new replies page.
We already have a button for that, you can simply unwatch topics you do not longer want to engage or get notified on anymore.

Eh? I thought that'll just remove the topic from "watchlist" page but we'll still get the updates of the topic on the "show new replies" page? Let me try that with this thread.
If I may add another possible benefit of what "ignore a topic" may offer, sometimes we want to leave a topic we're previously engaged on, either because the topic no longer interests us or it has turned into a meaningless discussion filled with spam posts. I am not sure there is currently a way to "remove" these threads from our "show new replies" page.
I have never tested it, but maybe if you deleted all your posts in those threads, it might work!? I am just guessing. If you have no posts there, no one can reply to those non-existing posts. Hence, you won't see those replies if you click the "Show new replies to your posts" button.
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That might work, but I think it'll be quite troublesome if I've already made a dozen posts on that thread, initially interested on the topic because the discussion is engaging, but later on it became a mega thread filled with spam. The ignore topic button will be a very much welcome shortcut.