Why can't we use normal Cox ph survival model by incorporating time-varying covariate features into it?

Using the stanford heart transplant dataset, I have seen articles on why time-varying covariates are important as an input for survival analysis models. But can't we incorporate the time at which heart transplant was done as a feature in to the normal Cox ph model instead of using time varying covariates? I didn't understand the need for incorporating time varying covariates. Why can't we incorporate the static value of if transplant is done, the time at which is done and if not done then -1.

For eg. for patient id 1 that received heart transplant at day 36 and died at day 39. and for patient id 2 that did not receive heart transplant, Why can't we train a normal Cox Ph model on this data like so ?

A value of -1 would literally mean that the transplant was done 1 day prior to day 1.

The representation of this dataset for analysis using time varying covariates would be as below:

id starttime stoptime transplant status
1         0       36          0      0
1        36       39          1      1
2         0        5          0      1


You're proposing the following model: $$h^{(i)}(t)=h_0(t)e^{\beta x_i},$$ where $$h^{(i)}(t)$$ is the hazard function for patient $$i$$ at time $$t$$, $$h_0(t)$$ is the baseline hazard function at time $$t$$, and $$x_i$$ is the transplant time of patient $$i$$ as you have defined it.

Under this model, $$e^{\beta d}$$ is the hazard ratio comparing the hazard function of a patients who received a transplant $$d$$ days apart. You've (implicitly) assumed that the hazard ratio comparing the hazard function of a patient who received a transplant at time $$t$$ to that of a patient who never received a transplant is $$e^{\beta(t+1)}$$. Another implication is that the hazard rate of a patient who received a transplant at time 0 is $$e^\beta$$ times that of a patient who didn't receive a transplant -- and the hazard rate of a patient who received a transplant at time $$t$$ is also $$e^\beta$$ times that of a patient who received a transplant at time $$t-1$$. Is that (very strong) assumption reasonable? I would guess no, according to the authors you mention.