Yes you can use the medi-gun to heal people individually and get your uber charge up, but you can save a lot of people's lives if for example a pyro charges at the group that your healing one-by-one and burns a lot of them, you might not be able to extinguish everyone before they die, the amputator will help quickly extinguish all your teammates at once leaving no one behind.
I personally love the amputator because it helps a lot of people especially when there are a lot of needy people who want you all to themself. Using the amputator taunt you can share the love.
Using the amputator and the crossbow gives you a set bonus (+1 health regenerated per second on wearer) which really helps you incase an enemy attacks you while you are taunting.
You can read more about the taunt here.
Here's a quick comparison of healing methods:
It allows you to heal multiple people at once.
Of course, the drawback is that you're open to attack when you do it.
The main benefit is on cp_degrootkeep, where you're not allowed to use your medi-guns.