I want to provide a bit of a counterpoint from personal experience:
A harsh firing process can actually even improve trust into management/leadership.
Poor performance especially in smaller companies is often very noticeable to coworkers and firing people for not performing at all can signal that management is willing to follow through with hard decisions for the good of the company (and that IS important to employees!)
If there is consensus in your team that your disappointing employee is useless, than firing him on the spot might be the best thing you can do to increase trust in leadership (but this is a hard decision and a lot more context would be needed to judge).
A harsh firing process can actually even improve trust into management/leadership.
Poor performance especially in smaller companies is often very noticeable to coworkers and firing people for not performing at all can signal that management is willing to follow through with hard decisions for the good of the company (and that IS important to employees!)
If there is consensus in your team that your disappointing employee is useless, than firing him on the spot might be the best thing you can do to increase trust in leadership (but this is a hard decision and a lot more context would be needed to judge).