Another take is to nurture a "tamagotchi" or "pet" phone (iPhone) at home, with some innocuous memes, following sports accounts and such. Bring that when crossing any border. Leave the "real" or "business" phone at home at all times. Of course, it's not a realistic solution for many people.
Bringing anything non-standard can be misinterpreted as, or worse, construed as something malicious.
The problem with an empty "burner phone" is, that can also look suspicious. Even if you have a receipt with you, they may wonder why you bought a new phone just for the trip. Lockdown mode seems even more suspicious.
Just don't stick out is unfortunately probably the best answer.
If you are entering a foreign country and they ask about your burner phone say that your normal phone is on a cheap plan that doesn’t support out of country service so you have a cheap burner for international travel.
If you are returning to your own country say it is because you didn't want to risk having your expensive phone stolen while abroad so you travel with a cheap burner.
In that situation, don't refer to it as a "burner". Using that word is automatically suspect.
Keep in mind that they hear plenty of stories like this. What it amounts to is that you're trying to achieve plausible deniability even if they assume you're lying, i.e. don't give them any concrete basis for suspicion.
Plausible deniability is probably the best answer. If you just tell them “no, I know my rights” if you get randomly selected, then you’ll get put on a list for being randomly selected every time. But if you appear to cooperate, you get let off entirely.
Bringing anything non-standard can be misinterpreted as, or worse, construed as something malicious.
The problem with an empty "burner phone" is, that can also look suspicious. Even if you have a receipt with you, they may wonder why you bought a new phone just for the trip. Lockdown mode seems even more suspicious.
Just don't stick out is unfortunately probably the best answer.