There are some countries like the UK where not talking to the cops can potentially land you in trouble too. (In England and Wales, juries are permitted to draw adverse inferences from a defendant not raising information with the police that they later rely on at trial.)
If you are arrested, exercise your right to counsel. This is guaranteed in both the US, the UK and most of Europe (under the European Convention of Human Rights), as well as lots of countries elsewhere.
Yes, it might mean you have to wait an hour or so for the lawyer to get there. Yes, the police will try to convince you that only guilty people need lawyers (not true) or that it'll make things worse for you (also not true).
The counterpoint to this is: if the cops are following procedures correctly and have legitimately collected evidence, they have nothing to fear from you exercising your right to free and independent legal advice.