You are overthinking it, but not in the way you fear. The real issue isn't whether to build a composite score or bin a "developed" dummy, it's that you're treating innovation as a single number when your own data suggests it's a system of interlocking signals. Patent applications alone don't capture how effectively a society converts ideas into productive output. Neither does social progress, work hours, or government type in isolation. What you're actually describing is a latent variable, and the honest way to handle that is not to force a composite but to let the data speak through a well-specified model that accounts for the confounders you already suspect.
Your instinct to add education, caloric intake, and infrastructure is sound, but the way you're framing the problem, normalize inputs against confounders, then build a score, introduces a layer of arbitrary decision-making that will quietly undermine your results. A composite index requires weighting choices, and those choices carry assumptions you haven't defended. Instead, consider a two-step approach: first, regress each innovation proxy (patents, social progress, work hours) on your confounders (education, calories, infrastructure) and extract the residuals. Those residuals represent the variation in each variable that is *not* explained by development. Then use those residuals as your independent variables in the GDP growth regression. This sidesteps the need for a composite entirely, keeps your categorical government type as a control, and directly answers the question "what is the marginal contribution of innovation once we account for baseline conditions?"
On your second concern, whether to add more variables, yes, you should, but not for the reason you think. Your current set is a weak proxy because patents measure invention, not diffusion or commercialization. Social progress captures well-being but not productivity. Work hours capture effort, not efficiency. What you're missing is a measure of *economic complexity* or *knowledge intensity*. The good news is you don't need a new dataset. Your existing infrastructure score and education levels, when combined with patents, can serve as a proxy for absorptive capacity, how well a country turns raw ideas into growth. Add those as interaction terms with patents, not as standalone variables. That way you're testing whether the effect of innovation on GDP growth depends on the enabling environment. That's a more interesting and defensible question than "does innovation matter?" because you already know the answer is yes. The real question is *under what conditions*.
Here's your concrete next step: stop building a composite. Run your OLS with the raw variables, extract residuals from a first-stage regression on confounders, then include those residuals in the second stage. Keep government type as a categorical control. Add an interaction between patents and education. If you're still worried about omitted variable bias, your caloric intake proxy is crude but usable, just be transparent about its limits. You'll have a cleaner story, a defensible method, and a finding that actually tells us something about policy: whether innovation drives growth directly, or whether it only works when the institutional and human capital groundwork is already laid. That's the analysis worth doing.