The calculation of the Cox regression coefficients depends on knowing the risk set at each event time. For convenience, I'm copying some relevant parts of this answer here for the situation without tied event times; $dN_i(s)=1$ represents an event at observation time $s$:
Differentiating the log partial likelihood with respect to the
parameter-value vector $\theta$ gives a score vector (Equation 3.4 of
Therneau and
Grambsch):
$$ U(\theta) = \sum_{i=1}^n \int_0^{\infty} \left[X_i(s) - \bar x(\theta,s)\right] dN_i(s) = \sum_{i=1}^n U_i(\theta)$$
where $X_i$ represents the covariate values for case $i$ and $\bar x$
is a risk-weighted mean of $X$ over observations at risk.* The
maximum partial likelihood estimator $\hat \theta$ solves:
$$\sum_{i=1}^n U_i(\hat \theta) = 0. $$
The critical point is that you need to know the composition and characteristics of the risk set at each event time in the data. If one individual's event occurs in the middle of another individual's interval-censored event time, then you don't know whether that second individual should be included in the risk set for the first individual's event.
...it seems that Cox could still work, but will probably not be very accurate, as it will ignore some data.
Ignoring those data is the point. Your scenario treats an interval-censored observation as irrelevant at the beginning of the interval, and treats it as no event at the end of the interval. Yet for an interval-censored observation, we know that an event occurred during that time interval. Replacing true events with non-events will not lead to a proper model.
A common work-around with interval-censored data is to assign the event time to one or the other end of the interval, typically the end of the interval. With the latter choice you have a correct model of the time that the event was observed rather than when it occurred.
That's often good enough for a particular application. For example, cancer recurrence times are usually interval censored, with no recurrence seen at one clinical visit while a recurrence was seen at the subsequent visit. So the recurrence occurred within that time interval. Although there are ways to incorporate interval censoring into a true partial likelihood calculation for a Cox model (D. Finkelstein, Biometrics 42: 845-854, 1986), modeling the observation time instead of the occurrence time is frequent practice.
*The risk score for case $i$ in a regression without case weights is $r_i(\theta,s) =\exp[\theta' X_i(s)]$. Then the risk-weighted covariate average is:
$$\bar x(\theta,s) = \frac{\sum Y_i(s) r_i(s)X_i(s)}{\sum Y_i(s) r_i(s)},
$$
where $Y_i(s)$ is the at-risk indicator for time $s$.
