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Gutter with or without a wing
Gutter Without Fascia
No! They call them Fascia Gutters because they can act as a fascia board. The reason for installing fascia boards is to cover the ends of the rafter tails, that are either 2x4's or 2x6's, which are part of the roof's structure. Fascia Gutters have a tall back, which allows the gutter to completely cover the ends of the rafter tails.

OGee Gutters and Half Round gutters are only 3-1/2 inches deep which exposes the ends of the rafter tails, which can then lead to dry-rot. These gutters should only be installed over fascia boards. Fascia Gutters can be installed with or without fascia boards.
Tape Measure in the Gap
Besides having the ends of the rafter tails exposed because the Ogee Gutter is too small to completely cover them, they also don't have a wing that completely covers the edge of the roof. Looking at the picture of an Ogee Gutter attached without fascia boards and since this house has a closed in overhang, you can see the potential damage from water gaining access behind the gutter.

Since this gutter is not even flashed, water is entering the house closed sofit during every rain. Even if this gutter was flashed it does not provide protection if the gutter was to become clogged. As the flashing is not sealed to the gutter. Having a gutter with a wing prevents any water from gaining access behind the gutter.
Clogged Gutters
gutter guard failure
The most common reasons gutters will fail are clogged drain holes, gutters without a wing installed without flashing, weak gutter support attachments, not enough downspouts or wrong downspout locations. When gutters become clogged with leaves, dirt or pine needles just about any type of gutter will fail no matter how it was installed. Installing gutter guards in areas that are under trees will help keep drain holes from clogging up. Also the size of the drain outlet used by the installer. A lot of gutter installers will use a 2 inch round outlet for a 2-inch x 3-inch downspout.

Action Rain Gutters uses a 2 x 3 punch which punches a hole in the bottom of the gutter thats the same size as the downspout. This allows more water and larger debris to pass through from the gutter to the downspout. When installing gutters without a wing, the gutter must be flashed with edge metal to protect the edge of the roof sheathing and keep water from splashing over the backside. Although this is not a water tight seal. If gutters without a wing fill with water, odds are water will be coming down the back of the gutter even if it is flashed. The only way to prevent this is by installing gutters with a wing.
Gutter with a Wing
No! All gutter styles don't play by the same rules. Fascia Gutters have a wing that lays on top of the roof plywood deck and under the roofing paper and shingles. This wing must lay as flat as possible as to not hold up the roofing. Since fascia gutters are also the fascia boards, they need to be straight and plum as possible with the roofline. It would be like installing new fascia boards with a slope. Your house would look like the foundation is sinking. Fascia Gutters should be installed with the roofline, just as if you were installing fascia boards.
Ogee Gutters, Half Round Gutters and Box Gutters do not come with a wing, so they can be hung at a slight pitch for drainage. Basically any gutter without a wing.
Copper Gutter with a wing
Add any slope using the gutter above would be impossible as that large wing must lay flat on the roofs deck. So obviously not all gutters must be installed with a slope. Fascia Gutters are only installed on the west coast. Which means the rest of the world most likely have never heard of them. Other states mostly install Ogee Gutters, which is most likely the gutter style the DIYer article is referencing too. Finding a website that talks about Fascia Gutters is rather difficult to find, but there are some. If the DIYers websites is talking about sloping gutters, then its a gutter without a wing.
Those who insist all gutters must be sloped for drainage, obviously have never worked with or installed Fascia Gutters. Back east most have never heard of a gutter with a wing. Although, this wing plays an important roll with the gutters function and rigidity. Not only does it prevent water from getting behind the gutter, this wing also helps keep the gutter attached to the house when under heavy water loads. Ogee gutters tend to pull away from the house over time due to lack of support, which is the whole purpose of adding a slope to this style of gutter.
Copper Gutter with a wing
Seamless Fascia Gutters are not a DIYer project. They are only installed by licensed gutter contractors, that have purchased a 20k machine. Contractors are not going to go online looking for help on installations. The whole purpose of adding a slope or pitch to a gutter is because in a heavy downpour, Ogee Gutters are small and can be overwhelmed by the amount of water. Adding a slope helps remove the water quicker. Fascia Gutters do not have this issue. Since there are less problems with Fascia Gutters, they are only installed by professionals and they are mostly only installed on the west coast, they are talked about a lot less on the internet.
Absolutely! Fascia Gutters that are installed level and plum with the roofline will drain just fine. There is normally a downspout installed every 30 feet to allow for drainage. The water amount that will remain in the gutters is no more than the water puddles left on the ground after it has rained. When the ground has dried up so do the gutters. If the homes roofline is not level and plum, downspout locations can be changed or added to the lowest point of the gutter run.
Over doing the slope of gutters
In the picture you can see an example of what happens when the installer gets a little crazy with the slope of the gutters. Gutter sloping only needs to be 1/4 inch per 10 feet. This gutter looks to be over 1 inch per 10 feet. It's sloped so much the flashing or drip edge is no longer covering the back of the gutter. A piece of flat stock will need to be installed under the roof flashing and into the gutter to prevent water from running behind the gutter. Or better yet, raise the gutter up to a 1/4 inch per 10 feet. The flashing installed in the picture is 1-1/2" x 1-1/2", if the gutter is all the way up under the flashing at one end and 1 inch lower than the flashing at the other, thats a 2-1/2 inch drop. Which is enough slope for a 100 foot run of gutter, 1 inch drop per 40 feet. Also notice the gutter continues to drop even after passing the downspout. Hope there is another downspout at the other end. To install the flat stock in this gutter, it will need to be notched around every hanger. Good luck with that! Another example of the pros of having a gutter with a wing, completely eliminates the potential of water getting behind the gutter.
Having a drainage system that keeps water away from the structures wood and foundation is the key. All gutter systems should function with the removal of water as to not be overrun its capabilities from whatever mother nature throws at it. When Ogee gutters are installed with a small 3 inch depth, having a good slope helps this small size gutter not to be overrun in a heavy downpour. Even if the wing is removed from the Fascia Gutter to allow a slope, when you increase the size of the rain gutter to a 5-1/4 inch Fascia Gutter any extreme slope will greatly amplify the negative look of the homes curb appeal, while only offering small improvements of water removal of the drainage system.

When using a larger Fascia Gutter which can hold twice the amount of water as an Ogee Gutter, the importance of gutter sloping is more overrated. Finding that midway point between function and appeal should be the installers main concern.