Should we let criminals choose their sentences?

The law allows the judges to choose sentences, up to a defined maximum sentence.

If someone is convicted of murder, why couldn’t the judge offer a choice of 3 possible sentences?

For example:
– the death penalty
– life with chance of parole
– 20 years with no chance for parole

If a convicted criminal chose their own sentence they relinquish the right to appeal. If they let the judge decide all stays as it is now.
They may not choose things like which prison. In fact, pretty much all other decisions but the sentence would be the charge of the court to decide. Death penalties would be enforced within the year, as there would be no messy appeals hang-ups.

What would be the pros and cons to offering a choice of sentence?