From your description it seems you have a need for dealing with unobserved heterogeneity (i.e. your 'latent ability issue'). Indeed, you can take this into account using a panel data model, such as a fixed effects model.
Propensity score matching (PSM) has the aim of estimating treatment effects or potential outcomes in case of discrete treatments. You propose to create this treatment variable using discretisation of your endogenous independent continuous variable. This is acceptable if the cut-off point at which you divide your sample this way is a meaningful quantity. In that case, all subjects on both sides of the cut-off have a fixed deviation in their outcome variable, i.e. there is a treatment effect, with naturally defined treaments.
Without knowing more about your study it is not possible to say whether there is an actual treatment here based on your continuous variable. If there is not, I would not recommend using this method, as you basically throw away information for the estimation of the treatment effect, which additionally is based on an arbitrary cut-off point. Also, estimating propensity scores is usually done when controlling for many variables, not just the one or few. With only a few you would be able to use matching directly on the variables, without any issues caused by high-dimensionality.
If your independent variable is endogeneous then the panel analysis may be enough to make up for this, provided the endogeneity is caused by omitted variable bias due to unobserved heterogeneity. In case there are other sources of unobserved heterogeneity then there are plenty of other options, such as instrumental variable analysis.