It depends on your phone's CSC code. Samsung's unlocked CSC code = BTU. Three's carrier locked code = H3G.
To check your CSC code, go to...
Settings > About phone > Software information > Service provider software version...
I too have found this to be problematic, i purchased a refurbished, unlocked S21 Ultra, put my Vodafone Sim in, but the origin Software still has the THREE logo on boot and shut down, the annoying thing is i presume i will be locked into updates for the phone only when Three release updates, (this may be the old method though).
Certainly should be able to disable the branding logo and Bloat ware as an owner of the phone, or it should switch with the Carriers Sim inserted.
my understanding now is that the S21 range don't allow for the original software to really be removed even after an Odin CSC firmware flash to the Samsung default.
If anyone comes across a work around (without a custom Rom) let me know... even if its replacing the Three logo with a black image.
Unfortunately not, even after going to a phone unlocking service, the three splash screen was locked in, i did read that there was a 'patched version' of an 'unlocker' that could flash the phone back to the baseline, but this isn't something i was prepared to do on the handset. Ironically.... the S21 Ultra recently passed away, and i purchased the S22 directly from Samsung... in a sense... an expensive way to get rid of the splash screen, but it worked i suppose. lol .