I don't know if you're looking for a phone that just doesn't go under the Google name and can run alrernate OS's, or if you're looking for a phone that's completely separate from Google, but Motorola and GrapheneOS are discussing plans to make it so GrapheneOS is compatible with Motorola phone, and it's estimated that something will come of it by next year. Other than that, at least one other person mentioned it, but your best bet is sadly a Pixel phone. For now, that is.

You're probably not going to like this, but I think a lot of your best options are Pixel phones of various kinds. A lot of the phones with unlocked bootloaders are indie phones that are quite expensive (largely because they don't have the benefit of economy of scale that, say, Samsung or LG or whoever enjoys) and the significant majority of regular consumer phones are really locked down and nearly impossible to replace the OS with anything not signed by the original manufacturer. But Google Pixel phones are some of the few counterexamples -- phones that have unlockable bootloaders despite being made by a big evil company.

I wouldn't recommend the stock firmwares on such phones, of course. But with a custom firmware, you can still "degoogle" as far as software is concerned.

Beyond that, be sure to do your homework. In particular, make sure that VoLTE is supported on the particular phone/OS combination you want to use or it won't be able to make voice calls.

LineageOS supports so many old phones, but no verified boot (bootloader relocking).
https://wiki.lineageos.org/devices/

I haven't tried it, but I used its predecessor CyanogenMod for years without issues.

.

/e/OS (a beginner-friendly derivative of LineageOS) also supports many old phones without verified boot. Some newer ones can verified-boot.
https://doc.e.foundation/devices/
The better supported ones are "official" builds while the others are "community".
https://doc.e.foundation/devices/?channel=official

I've been using /e/OS community release on a Sony phone for 1.5 years. It's okay. Most apps work with MicroG. Bootloader not lockable, but apps don't notice.

Some bugs:

• Locked-screen camera lauch long-pressing the shutter button doesn't work (have to unlock first)
• The arrow on the screen pointing at the side-mounted fingerprint reader points at the wrong place
• The setting about never rotating the home screen when turning the phone on its side doesn't work, it just keeps rotating

https://www.quora.com/How-does-Murenas-custom-e-OS-compare-to-Googles-Android/answer/Harri-K-Hiltunen

.

iodéOS, another LineageOS derivative, has verified boot on some devices:
https://iode.tech/iodeos-official-supported-devices/

I haven't tried it.

Depends on where you're based for the price. But fairphone 6 is cheap and you can buy it with a custom OS straight from the website - e/OS, which I believe is derived from LineageOS. The e/OS version is £570.

That's one that I'm using. It's great, I've had no problems with it and, if anyone wants to, my banking app works good.

go to gsmarena and filter phones by chipset. select something like Snapdragon 845, a widely supported SoC that is at the cross-section of hella old i.e. cheap but also super capable; it's doubtful you'll ever need anything faster than that. then see which of the filtered models you can get for cheap. finally go over to lineageOS and see if it's got current support.

I think the pixel 6 loses support this month but, 7s and up are relatively cheap used and can use graphene OS. (Comment posted from pixel 8 running Graphene OS)

My first experience with a custom ROM was the cheapest pixel 3 for I think $80 AUD back in 2021 and grapheneOS.

Just went past every 2nd hand store/pawn shop every weekend untill I found what I needed.

CalyxOS supports some Motorolas:
https://calyxos.org/install/

I haven't tried it.

Nuru? They renamed postmarketOS just a few days ago, can't remember how's called now. But it's compatible with non-flagship devices. Else you're into other OSes that are trying to be competitive in this ecocidal market ;•\

Not sure if Sony Xperia 10 III is "cheap" yet, but it can run Jolla's Sailfish OS Linux.

I haven't tried a Jolla Phone.