Yes, but if you wanted an attorney present for questioning it would take away that right. Yes, everyone gets a lawyer in court, but they should also have one during questioning if they so desire. Why do you think it’s a good idea to overturn it? Best approach if you are arrested is to stay completely mute except for "I wish my attorney present", "I need to go to the rest room", and "I'm hungry/thirsty. I wish food/drink".
Nothing not on the above list should come out of your mouth.
I'm trying to find this video I saw a few months back about a former prosecutor stating that you should always remain silent.