Would this be an appropriate way of handling internal time-dependent covariates when fitting a survival model? By definition, internal covariates 'exist' or can be recorded as long as the study participant is alive.  Let's say we have an internal covariate that is continuous in nature.
Given that, would splitting at failure times and using restricted cubic spline to model this time-dependent internal covariate be appropriate?

In principle, yes. Under the Cox model only the covariate values at event times enter the analysis. Thus you only need to specify their values at event times. Flexible modeling of continuous covariate values, as with regression splines, is a good idea if you can devote enough degrees of freedom to them.
A danger with time-dependent covariates is that you might introduce survivorship bias. You need to apply your knowledge of the subject matter to make sure that isn't the case.
The time-dependence vignette of the R survival package provides extensive advice on how to handle these situations and explains tools that can help format data appropriately.
