In all these years ARC has been in play, it has never achieved universal "just works" quality: this device connected to that device doesn't work well but one of those connected to this other thing does. It doesn't even offer simple troubleshooting tools to help one figure out how to resolve such issues. Often, it's some kind of process-based configuration tests to maybe eventually find a sequence of connections that gets you what you want.
My opinion is to seek other ways to replicate what you want it to do rather than rely on ARC. For example, a universal remote with macros can generally make any home electronic setup work as people want ARC to work. The remote becomes the ARC-replacement "brains" of the setup and the macro lets YOU manage the links in the HE chain.
However, if you want to try to find some way, work through the order of things being turned on. In other words, mix up that order to see if you find a magic combination that gets you what you want. In playing with ARC in my own setup, I came to discover that if I let AppleTV be the FIRST ON device, the receiver and then TV in the chain work pretty well. Similarly TV as FIRST ON will turn on the receiver and work well (but won't turn on AppleTV). Receiver on first won't light either, nor know which mode I might want it in or input I want to be in control.
OP, in your situation, AppleTV has to "handshake" with those HomePods. There's a little time there to negotiate that connection each time. AppleTV does NOT assume your Homepods are still available to it each time you turn it on. So it needs to scan for them and connect each time.
So ARC is probably "searching" for speakers when turned on and not "seeing" Homepods (yet), nor any other, so it "thinks" it should use TV speakers. Then AppleTV succeeds at fully connecting with Homepods, reports that these other speakers will be used and the TV's ARC then bows out the TV speakers to use the HomePod speakers.
Since there is always going to be a bit of time for AppleTV and Homepods to fully connect, I don't think you'll get a simple ARC "ON" process. If there was some way to delay when the TV turns on for the time it takes for AppleTV and Homepods to fully connect with each other, that would likely do it. So this would be: AppleTV on... AppleTV locates & syncs with HomePod(s)... THEN TV on... ARC "sees" that audio has somewhere to play other than the TVs speakers so it doesn't need to assign the TV speakers.
In my own setup, I have WIRED surround sound speakers via a receiver. So using ARC, AppleTV on then "calls" for Receiver to turn on. Receiver needs no time to determine what speakers are attached because they are wired. Receiver "calls" for TV to turn on. TV doesn't need to turn on TV speaker because ARC has assigned Receiver for audio already. Thus, I get what I think you seek: a simple, everything "ON" without things blinking on and off as the system "figures out" where video and audio will be played.
Probably the simplest solution for you would be to abandon ARC and use AppleTV remote to control AppleTV + Homepods part... and the TV remote for TV on/off.
OR buy a macro-based universal remote. This would get you one-button-click, everything-on and off functionality. In this setup, without ARC trying to find speakers not yet acknowledged by AppleTV, it would simply use the TV as a video monitor. AppleTV and Homepods would sync and work as you wish. Downside to this: still 2 remotes if you want to use the AppleTV remote/SIRI.
BUT, before you go either way, play with the order of when you turn things on. For example, if right now you are turning on TV first then AppleTV, flip that order. If TV is first on, there's nothing else for ARC to "see" so it assumes TV video and audio. Then, you turn AppleTV on, Apple TV executes the process to find and connect with HomePod(s) and then "tells" TV that audio should be shifted to them. Net effect: TV on, TV speakers on, AppleTV on, short delay, Homepods found & connected, TV speakers off/ignored.
In the longer run, consider putting a Receiver in the middle with wired speakers. Homepods were never intended to be used for any true TV surround sound... and real surround is better than even HomePod stereo. If you go there and have the budget, I suggest not trying to shortcut it with the soundbar route either. 5 or 7 speakers placed correctly is much better than trying to fake it with one bar- even those spun as ATMOS bars- front and center. Faux surround is not surround. Faux ATMOS is not ATMOS. Your ears would hear the difference if you compared the two.