To spread the ink smoothely and evenly, it is BEST to use a flexible squeegee, wide enough to cover the image in one or two strokes. You can get these squeegees at the same arts and craft stores that sell silk screen equipment and screens.
For smaller screens, you MIGHT be able to get away with something like a windshield wiper blade. But the rubber part is a LOT thinner than a silk screen squeegee, you'd have to be VERY careful not to tear the screen as you wiped. With printing on paper, you can get away with a faily light but steady pressure. For printing on fabrics, you have to bear down a little harder to make sure the ink penetrates the fabric's fibers.
If you are only making one or two prints, you might get away with using a piece of cardboard, cut with a perfectly straight edge. The ink will be absorbed and soften the cardboard, so you won't be able to use it for many wipes.
When I did silk screening many years back with a vacuum screener, all I had was a squeegee to push the ink around... Scoop coater... I've never even heard of one.
The printing I was doing had to be extremely accurate (checked with a low power microscope), and I managed just fine with a squeegee.