In a nutshell, we can stay silent so who cares?
The problem with that is that everyone should care. By the nature of their jobs, police deal with people with a variety of disabilities, levels of education and language comprehension. I don’t know that it’s reasonable to expect people to have the presence to remain silent. Yet if they don’t, we’ll happily incarcerate them. I don’t like crime but I hate innocent people being punished for crimes they didn’t commit.
Likewise, say it happens on a Monday. Your work is gonna wanna know where you are. They can definitely get away with firing you simply for getting arrested due to "At Will Employment" since you can't really go to work while in a jail cell.
While pleading the 5th protects you from the government forcing confessions or something similar, it doesn't save your job.
When they lie, those statements are constructed to pull informations out that would otherwise be hidden. If the lie is constructed with leading facts it might end up lead into false confessions (thats bad).
Investigator can include false info during the 'lie' to try to spot a false confession (covering for someone else, or even confusion and stress and wanting to get out of hostile environment).
Bad investigator will lie and force a confession facts onto suspect. As already mentioned some people might crack under stress of the situation and go into fight or flight - 'get me out of here I will tell you anything' - because they were fed actual crime facts, suspect will use those to construct a confession that will look like a real thing. That practice is something people should get fired over.