Originally I was fairly swayed by the OP. Then lots of posts pointed out the whole"shades of gray" thing in terms of the legal system not being black and white. Easy to forget for us programmers. Still others pointed out downward spirals like the Salem witch trials. Hmm, perhaps a weaker point, but still good. I get the feeling the OP is still not convinced, however. So here are a couple of simple things I thought of after reflection.
Not in the form of a scenario, so fail0 me if you wish, but in this case I think clarity is not best served by a rambling scenario.
Basic premise: people so not ask simple questions that are yes/no when trying to extract information. They ask questions that are convoluted and complex, trying to catch someone in a lie. This is fine on tv, and acceptable when the party is guilty.
But you can't know the party is guilty. Many people get triples up under stress. Many are not articulate. Many (by definition) have below average intelligence. A goodly number do not speak English as well as the general public would hope. A goodly number have hearing problems, mental illness, senility, or other issues that prevent coherence under time-pressing pressure situations. There are even people, believe it or not, who have not grown up in a hostile or debate-heavy environment, and thus may fall into all sorts of verbal traps simply out of being an ingenue (or the male equivalent).
Oh, that reminds me: the point someone else made about all of the crazy statutes around, plus the vague nature of many of them, means that all of us, no exceptions, are lawbreakers. If that sounds silly or melodramatic, I'm not sure why. Look up crazy laws on the Internet, they are funny and concerning both. Also not known to the public or law enforcement in general, and thus not enforced.
Which really brings me to my main point, which is the need for counsel: many people don't know all of the rules of the "game" they have just been thrust into when they are entering the legal system. Many are petrified by fear and will make egregious mistakes. All are in violation of frivolous laws not material to the case. Potentially a majority are likely to be caught in verbal traps laid by experienced interrogators. Forget torture, it's hardly needed.
As for benefitting the guilty more often than the innocent: NO. This actually benefits the innocent more than the guilty (on a case by case basis, not in overall percentages that include the disproportionate number of guilty interrogatees). Repeat offenders are more familiar with the system. So are those from a culture of crime. Those with high intelligence and a taste for crime have an excellent incentive to study up. Those who come from hostile environments (not all of which are crime prone, but certainly many are) may be used to hostility and this on average less likely to get shaken and thus fall into certain kinds of verbal traps.
In other words, the right to remain silent is the same as the right to have counsel speak for you for many practical purposes.