If you have to pump the lever to get it to work, it sounds like air in the system so it should be bled.
I doubt the air was from pushing your pads back a bit.
I wouldn’t call it ‘easy’ to do. But it is quite a methodical routine. If you follow the instructions, it will work and give good results. Things to watch out for are:
~ contaminating pads with brake fluid – when you attach/detach the syringes small amounts of fluid can leak out. Just make sure you’ve got a rag to stop it dripping onto pads
~ being too heavy-handed with the syringes – you don’t need to really force them. If you do you might squirt fluid past the connectors (or draw air in)
That looks like the cheapest bleed kit on-line.
you might as well do both ends while you’ve got the kit out.