The house has chosen the new speaker. He is an interim because of the agreement of the house to hold another vote at some point. But he is for all intents and purposes the new speaker. The house members voted to replace the former speaker with a temporary one, that is how they chose. Additionally the voting system can be anything the house wants it to be. If the house says that the current speaker gets to choose the new speaker, that would fulfill the requirement. There is not specific method the house has to use to choose the new speaker.

They could have foot race, and the winner of that foot race is the new speaker. Totally constitutionally legal. They can also choose a new speaker in any manner they wish.

As far as how the current speaker of the house was chosen. From the public timeline of the The House (reverse chronological order):

04:46:55 PM The House received a message from the Acting Clerk. The Acting Clerk notified the House that the Honorable Patrick T. McHenry is the first name on the letter he received pursuant to clause 8(b)(3)(B) of rule I.

04:46:33 PM The Chair announced that the Office of the Speaker of the U.S. House of Representatives is hereby declared vacant.

04:45:30 PM H. Res. 757 On agreeing to the resolution Agreed to by the Yeas and Nays: 216 - 210 (Roll no. 519).

https://ethics.house.gov/sites/ethics.house.gov/files/documents/117-House-Rules-Clerk.pdf clause 8(b)(3)(B) of rule I

“(A) In the case of a vacancy in the Office of Speaker, the next Member on the list described in subdivision (B) shall act as Speaker pro tempore until the election of a Speaker or a Speaker pro tempore. Pending such election the Member acting as Speaker pro tempore may exercise such authorities of the Office of Speaker as may be necessary and appropriate to that end. (B) As soon as practicable after the election of the Speaker and whenever appropriate thereafter, the Speaker shall deliver to the Clerk a list of Members in the order in which each shall act as Speaker pro tempore under subdivision”

So to translate. According to the rules of the current session of congress (117th) that was ratified by the house of representatives after they were seated in January 2022. If the Speaker has been declared vacant by the majority of representatives, then Rule 1 Section 8 will be invoked. So that list that “some guy” made, is the actual and constitutional next speaker of the house. End of story. There is nothing magic about it. It was just the agreed upon rules of this congress from 10 months ago.