It's an intriguing concept. But it's harder than you think. It's not only an engineering problem, but a manufacturing one too.
I think your best bet is to build a standard pre-engineered truss design that already has the strength to meet code. And then, use the sheeting for added strength. That would take the engineering out of it. There's still manufacturing. By contrast, the guy in the link is using square trusses, which is the worst truss design possible. He is betting the farm on the strength of the sheet material.
Dimensional lumber is not straight. Even with well-chosen boards, you'll need to run it throught a jointer edgewise to straighten one edge, then through a planer to straighten the other edge and bring it to your standard width. Since your standard width must be the lowest common denominator of all the boards you can get, be prepared for this to gobble as much as an inch / 25mm. So you'd need to start with wider boards.
The good news is, with such a narrow truss, you could do a minimal jointing to get the long members straight on one side, then assemble the truss, then run the assembled truss through the jointer and planer to square it as an assembly.
To assemble the truss, you'll need to lay them all down on a flat reference table (where?) and frame them up. I would construct the truss members to fine tolerance, and screw and glue to maximize the strength of the truss without sheeting.
Then plane the second side of the whole truss, flip it over and give the first side a last pass through the planer to correct any irregularities. Gluing the sheet needs to be near perfect. And the manufacturing technique needs to be consistent.
Nails and screws will have no strength when affixing the sheet material. All the strength of the sheet material depends on the glue. This is no place for random hardware-store glue. I would probably go for an epoxy if I could keep it out of direct sun.
The people in the link used nails, but they used an awful lot of nails, and they are dealing in much taller beam sections than you are. They also said "glue it too", I am saying "glue it well". For you, space matters so precision and strength matter.
It's important the wood be fully dried when working with glues. It takes weeks to stabilize kiln dried lumber bought off the lot, and months to stabilize wet lumber.
Late to the party, and I hope your patio turned out well.
For anyone finding this question in the future, the answer could be a 4x10 beam with 14' spacing and a patio span of 15' 2x8 rafters.
https://www.escondido.org/Data/Sites/1/media/pdfs/Building/InfoGuideline8b.pdf
A lot of cities or counties will have standard drawings for simple structures, and you can check building departments for things like live load and dead load requirements (snow and wind). This one seems to be a good example that includes span tables and connection details.
Best Answer
Forte Web is an excellent tool for designing wood beams.
That being said, it can take a point live load at mid-span of 610 pounds. Make sure you purchase #1 or select structural. You don't want any knots in the beam because of the types of forces a swing will apply to it.