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Viewed on a systemic level, everyone acting this way would give false-accusers a great deal of power.
This does not make it right, and any system where this holds true should be either abolished or fixed.
you could easily fall into the trap of false confession, and the real criminal could possibly get away with it.
It's up to the prosecutor as to what charges will be brought against you. Being open and honest could implicate you with even more charges. The best policy is to say nothing.
One should never lie on a court.
This is an advice a lawyer will give you, simply because the evidence against you is too overwhelming, and fighting for your innocence will most likely result in a harsher sentence.
Your assumption is that you will know the final outcome of the consequences. There is no way to fully know the future repercussions of admitting to a crime which you did not commit. Since committing a crime is a known negative value, the rational course would be to not confess to a false charge.
However, if you do so, is it acceptable to deny the offense in other settings?