In longitudinal data, what can be gained from modelling "time" as a covariate? Imagine some data set where subjects $1, ..., K$ had a response $Y$ measured repeatedly over a period of time (say, 2000-2015). 
Now, in order to study effects like "does the month of the measurement matter?", or "is there a trend over the years?", or "does the day of the week matter?", we can of course use these variables (month, year, weekday) as covariates with fixed effects. For example, "month" and "weekday" could be a factor, and "year" could be a continuous covariate with some slope parameter to be estimated.
However, online, I have found many examples of people suggesting that we also include a time $t$ variable which measures the continuous change in time, i.e., "days passed since the start of the experiment". 
My question is, why? What is the use of such a variable? If there are some time effects, then they are going to be captured by the aforementioned fixed effects, no? 
So what would be the point of complicating things by introducing a "days"-variable, which presumably does not matter too much in the determination of the response variable $Y$, and which also is not going to be a "nice" covariate, in the sense that there are probably going to be many days which has no responses (e.g., entire months during the 15 year period where the experiment was on a break, usually during the summer period). 
 A: There are data sets that require/can use deterministic trends using the counting numbers perhaps having multiple trends. There are other series (more frequent in my long experience) that require differencing and one or more steady state constants to deal with multiple trends. Only the data knows for sure which approach is appropriate which is why I favor software that tries both types and assesses the dominant approach. Of course empirically identified level shifts also need to be investigated as they often occlude trend detection.
There is an indeed an argument for day of the month variables as they can capture systematic activity for different days of the month such as the 1st , the 15th etc. To continue this thought we have often found series that a week-in-the-month variable was important suggesting identifiable within-month struture.
Humans and nature often act in systematic patterns and extracting that minimally sufficient structure to decompose the observed series to signal and noise without over-fitting. 
All of which I speak ( lead and lag effects around holidays etc ) require one to identify and down-weight unusual values otherwise they cloud the model identification process
