this might be a misunderstanding on the purpose of the Police. If you are being interrogated, they are trying to put you in jail until enough reason is given to not. They are allowed to outright lie to make this happen, so long as you divulge something to them.
This is one of the inherent problems in the "right to remain silent". you are correct in identifying that silence (especially in contracts) is an acceptance in dishonor. Furthermore, by going silent, you are instead accepting a limited subset of rights, in lieu your inherent rights guaranteed by your constitution (in North America anyway), which includes waiving your rights guaranteed by the 4th amendment and 5th amendment.
by saying "yes" to the question "do you understand these rights and they have been read to you", you have given up your right to remain silent, which means you've now waived your original rights, as well as the new rights they read you.
Knowing your rights usually leads to a harder/harsher grilling by the police, as they attempt to test you on them. Not knowing your rights leads to an attempt to railroad you through the system.
The safest path has been "Duress" as implied by Uniform Commercial Code Article 1 Section 308. Inform the police in every encounter (i.e. every time they enter or leave the room) that you are acting under duress, and you intend to cooperate fully. Answer all their questions without hesitation, maybe adding a "please don't hurt me" to the end. If you're given anything to sign, your name now becomes "Without Prejudice" or "Under Duress", so sign it that way. Print that name underneath as well. Once released from police custody, you have 72 hours to inform them that your signature was given under duress, and that you're revoking those agreements/documents signed.