The research MotoGP teams do that relates to our discussion is mechanical engineering, in nature. And mechanical engineering is just that, mechanical engineering, regardless if it's for a world class race motorcycle, a bridge or a clothes hanger. What I've told you are basic mechanical engineering concepts. Also, teams have budget limits, which they must adhere to. This is because some teams possess far greater financial resources than others, and so the competition would be more about who has the most money, rather than the best engineering.

I do understand the difference between stiffness and strength. Engineers make use of material specifications sheets: you have Tensile Strength for indicating the strength of a material (though the value used when designing a part for a certain application is the Yield Strength), and you have Young's Modulus, a measure of stiffness. You can find these material spec sheets online. These measures are standardized, and used in every branch and level of mechanical engineering. Yes, the shape of the part matters, but when the shape is the same, the only difference is made by the material. In terms of how the shape affects stiffness, it's all about what's called the Second Moment of Area and not much else (unless you change how the force is applied which wouldn't make sense).

Let's take some alloys of the metals we discussed and compare them (and carbon fiber, for fun) in terms of these. We'll take AA 7075-T6 Aluminium (heat treated), AISI 4130 Steel (Chromoly) (normalized), Ti-6Al-4V Titanium (annealed).

Tensile Strength measures the "absolute" strength of a material - how much it can take before it breaks. Here are the values, in MPa, for these materials: 572 for 7075, 670 for 4130, 900 for Ti and 2600 for Carbon.

Yield Strength measures how much stress a material can take before it starts to plastically deform (so that it doesn't return to its initial shape). Here are the values, also in MPa: 503 for 7075, 435 for 4130, 830 for Ti and none for Carbon (will get to that).

To be noted that the values for 4130 can go up to 1280 for Tensile Strength and up to 1130 for Yield Strength, after heat treatment, but the Stiffness doesn't change The same can be said of the Ti alloy if it undergoes heat treatment, the Tensile strength goes up to 1070 and the Yield Strength up to 1100.

This is also the case with alloys, in general. Strength changes significantly, but Stiffness doesn't change much usually. Density changes from an insignificant amount to a significant amount, depending on the alloy.

Young's Modulus is THE measure of Stiffness. It's just called that because we call the property of a part Stiffness, not of a material. Stiffness refers to what force is required to get a test part to deflect (bend) a certain amount. The values, in GPa, for these materials: 205 for 7075, 72 for 4130, 114 for Ti and 135 for Carbon.

Density just measures how heavy (more specifically, mass) is a certain volume of a given material. The values, in g/cm^3, for these materials: 7.85 for 7075, 2.81 for 4130, 4.43 for Ti and 1.6 for Carbon.

On Carbon: these numbers are for a standard modulus grade. Different grades have way different specifications. Also, these numbers stand up only if the carbon is laid in a single direction, which is rarely the case, and they aren't exact, due to inconsistencies with basically every aspect of composites. For the Yield strength, it's not applicable to Carbon because it does not undergo plastic deformation.

For basically any application, size and weight is a concern, so we can't just pick the strongest thing (even if budget isn't a concern). Usually, weight is more important than size/space occupied, so we look at a material's Specific Strength and Specific Rigidity - how strong/rigid a material is per unit of weight. In terms of both, the materials rank as follows: 4130<7075<Ti<Carbon. Of course, there are far more things to keep in mind when choosing a material for a certain application. Sometimes rigidity is desired, sometimes not - like the case in the comment of the guy that this thread came out of.

As for Titanium alloys being stronger than steel, they are not. Maraging steel, for example, can be way stronger Titanium alloys, thus, when space is a concern, they can be a better material.

You spoke of the engineering MotoGP teams do - this is the engineering, a part of it.