The spell research guidelines aren’t really a capability of anyone, spellcaster or otherwise. The entire concept of researching a new spell is entirely up to the DM—the guidelines just provide ideas for a DM who wishes to make a character work for it in-character. But the actual ability to get a new spell comes from the Rule 0—that is the DM changing the game by introducing new or modified content (allowing a spell that doesn’t otherwise exist or isn’t otherwise available to the spellcaster in question). If the DM is for it, then it doesn’t really matter what the guidelines say, and if the DM is against it, it really doesn’t matter what the guidelines say.
In short, the guidelines are there for DMs, not for players. They provide a way of explaining an unusual spell in-character, and they allow the DM to breathe life into the concept of a spell researcher.
So the question of whether or not anyone “can” research a new spell is entirely up to the DM; I wouldn’t even say that a wizard “can” do so unless we’re talking about my table. At someone else’s table, my answer to that question would be no good—whether or not a wizard at that table can is up to that table’s DM. Equally, whether or not a warlock can research a spell, or an invocation, is up to the warlock’s DM, and what it means for a warlock to research a spell is likewise.
For what it’s worth, warlocks qualify for Precocious Apprentice, which allows them to prepare and cast a single 2nd-level spell. A lot of DMs aren’t going to allow it—it certainly reads like an oversight—but RAW, it’s a thing. That would make a warlock into a true spellcaster, and if allowed to research another 2nd-level spell... well, who knows how that would interact with Precocious Apprentice. It certainly isn’t clear. Again, up to the DM.
