There can be confusion between time-varying covariates and time-varying regression coefficients.
If a covariate’s value changes over time while its instantaneous association with hazard is constant over time, you can model that with the “counting process” data format. In that case, you can get a single estimate of its regression coefficient (log hazard ratio) that doesn’t change over time. In principle, the proportional hazards (PH) assumption can still hold.
Sometimes, however, the association of a covariate’s value with hazard changes substantially over time (even if its value stays constant). That means the PH assumption doesn’t hold. There then are ways to estimate how that association changes over time.
The time-dependence vignette explains the above statements in more detail.
One way to handle time-varying coefficients is as you describe, breaking time down into separate intervals and modeling different coefficients for each time interval. In that case, however, there is no longer a single hazard ratio that describes the situation for all times. It doesn’t then make much sense to get a “weighted average” over the time intervals, as what you would then be doing is throwing away the more detailed information about changes over time that you worked so hard to obtain.
If you are willing to work with some type of average hazard ratio, even if PH doesn’t hold a Cox model still provides a type of event-weighted average of the regression coefficient, which you could then exponentiate to put into hazard-ratio terms. If you combine that with robust standard errors you can even perform inference with the results, for example testing whether that event-averaged coefficient differs from 0 (hazard ratio differs from 1). See this page.
