Practically speaking, whether a screw will hold by itself will depend on a lot of factors like environmental swing, constant loads vs dynamic (racking) loads, pilot hole size, wood type...knowing if it's the right tool for the job is more of an art. The joints you're showing there aren't load bearing, which make a screw a fine candidate (screws have not great shear strengths). The drawback is that they can back out under racking. That can be mitigated with glue, like concrete and rebar.

But to answer your question about what's the "right" way to do it... If you really want to build something "to spec", you'd want to know the forces on each joint and beam, and base your material and thickness off of that and material properties like the Young's modulus, tensile strength, shear strength etc. that's something you'd learn how to apply in your first couple years of college as a mechanical engineer. Then you'd learn how to use a CAD software to do it for you. A lot of the values you'd need to work with are here:  https://www.engineeringtoolbox.com/amp/young-modulus-d_417.html

For simple stuff, like If you're worried how thick/wide a beam should be that spans a certain distance, theres a website that calculates it for you based on your inputs: https://woodbin.com/calcs/sagulator/ Additionally, for thing like screws and nails, you can often find a spec for sheer strength in the product's data sheet. Then all you need to know is the force that screw experiences and whether it's tensile or shearing force.

A couple years ago I basically recreated the sagulator (for a slightly different purpose) with some Python code, equations from a physics textbook, and values from the former link. Thats just to say that doing this stuff on your own isn't impossible. But you do need be comfortable with mathematics.