My point with the "but" was that the reason why lawyers advise this is not because police are trying to harm you. It's because in general, people are very bad at law and providing testimony. So maybe I should've clarified that -- the police are not at fault for doubting flawed testimony.
To be fair, police do engage in manipulative tactics in questioning, but that's kinda their job. The job of the person being interrogated is to ask for your lawyer and say nothing else.
It's good to keep in mind how police are incentivized. They are paid to arrest suspects and close cases. Whether or not they nab the wrong person hardly matters to them. So although they generally may not intend to harm you they will gladly do so if it increases the numbers on which they are graded. And if you happen to have some cash on you when you make the mistake of talking to them then there is the bonus opportunity of asset forfeiture.
To be fair, police do engage in manipulative tactics in questioning, but that's kinda their job. The job of the person being interrogated is to ask for your lawyer and say nothing else.