Yes, you can consider a discrete-time model here. Interval censoring means that you know an event happened within some time interval, but you don’t know exactly when. That’s how you treat event times in discrete-time models.
Right censoring is when you only have a lower limit to an event time, because of dropout or the administrative end of follow up. Individuals with right-censored times are removed from analysis after you no longer have information about their event times. In a discrete-time model you need to think about the interval at which you cease including an individual with a right-censored time in the analysis. Those who complete the full study follow up without an event should be included in all time intervals. Those who drop out part-way should be removed after the last interval during which you know that you had information.
In response to edited question and comments
Don't forget the importance of having a clear definition of the time = 0 reference time for the survival analysis. Statements of time values or time intervals are then all with respect to that reference time. If the reference time is the date of surgery, and the times intervals between visits are the same for all individuals, then the time intervals relative to the reference are the same for purposes of discrete-time survival analysis even if the calendar dates differ.
If you have individuals who miss some follow-up visits and then come back with an event, you have wider spreads of interval censoring for those individuals. You might be better off using models that can handle arbitrary interval censoring, like those provided by the icenReg package, instead of discrete-time analysis.
Now that you've provided more information about your study, it's clear that it requires a lot of care in thinking about and setting up the data beyond these issues of interval censoring of event times. One big question is the time interval in which you should first include a change in cognitive status that was identified at a follow-up visit. You don't know when, during the interval since the prior visit, that change in status occurred. So do you mark cognitive function during the time interval prior to identifying the change in status as YES or NO?
Also, I question the binary treatment of cognitive function as a predictor. There are graded measures of cognitive function, like the Montreal scale; an arbitrary cutoff of a graded score is not typically a wise choice. Furthermore, a graded score would be more amenable to the type of joint modeling of covariates and events that might handle this situation better.
I'm glad to have been able to provide help up to this point to clarify the principles involved, but you should get some experienced local statistical advice before you proceed with designing and performing this study. The issues are too complicated and specific to the field of study to provide detailed help on a site like this.