I wouldn't be surprised in the least to see Gabbard stage an independent run for the presidency in '24. Gabbard is all about self-promotion and trying her hardest to get someone, anybody, to listen to her. Therefore, an independent run for the presidency makes sense.
I don't know if the Libertarian Party would have her. Perhaps. No one else really wants her.
I still think Gabbard would prefer having her own show on Fox News or somewhere else.
No way Gabbard would receive 10% of the vote in a general election.
As for who Gabbard would hurt the most, my guess would be the Republican candidate, if she could actually hurt anyone. I don't know of any voting Democrat who likes or would support Gabbard. On all the Democratic boards I visit, Gabbard is mocked and routinely dismissed (the very few times she is even mentioned nowadays). Democrats don't like her and have no use for her. Even the few progressives who tried to defend her at one time do so no more. She doesn't appeal to Democrats any longer. Democrats are not worried at all about Gabbard.
Therefore, I would guess most of her limited vote would come from those more inclined to support a Republican or an independent candidate. Disgruntled Republicans wanting to send a message and some independent voters (who aren't going to vote for the Democrat or Republicans anyways).
Speaking of a independent or third-party run in 2024, I would keep my eye more on Trump than someone like Gabbard. If Trump runs for the '24 Republican nomination, and loses, would Trump be able to accept this? Would Trump admit defeat and get out of the race or would he claim the nomination was stolen by RINOs and decide to run as an independent candidate? This is something to keep an eye on in my opinion, more so than what some washed-up politician appearing on Fox News is going to do.