An even better token of respect is to just can the person and give them a respectable severance package, so they have months (plural) to find their next thing.
PIP also doesn't prevent someone from getting severance, firing is firing.
2. People, even lousy workers, talk. Treating people with respect on their way out may decrease the person’s negativity/bitterness. Consider it a marketing expense.
3. Severance packages come with legal releases attached, at least in the US. Less risk of lawsuits, frivolous or otherwise.
The length of employment is the most important part of the equation.