[…] de-trend and de-seasonalize a time series first so that it becomes stationary.
First of all, this does not necessarily make the time series stationary. There are many other forms of instationarities that you will not remove this way.
Aren't the trend and the seasonality of the time series the more relevant pieces of information?
Up to this point, you are right. If you have a detectable trend or seasonality in your time series, these allow for a first forecast and any more sophisticated forecast will have to build on these. I assume that the authors of those text consider this forecast so trivial that they forgot to remark on this fact.
One those two are removed isn't what's left a random process equivalent to a coin toss or throw of a die for each new observation?
When one removes the trend seasonality, is there any deterministic signal left to forecast?
No, there are many stationary processes (i.e., without trends and seasonality) that have some memory and thus allow for a sophisticated prediction. This includes deterministic processes (deterministic dynamical systems), but also stochastic processes with some memory, such as Markov chains / autoregressive processes – you do not need determinism to forecast, you just need some memory.
As already mentioned, you may also have some more subtle instationarities in your de-trended and de-seasonalised time series (the residual) which are distinct from a memoryless random process and that allow for some forecast.
The same applies for the aforementioned other forms of stationarities
Most introductory texts or tutorials to time series forecasting mention that one should de-trend and de-seasonalize a time series first […]. It is then easier to forecast future values of the time series.
The idea behind this is that once you are aware of a trend or seasonality, you can and should use it to forecast, but they also obfuscate features that can be used for a more sophisticated forecast. Thus you remove them, try to forecast the remaining time series (the residual), and then combine these forecast of the residual with your trends and seasonalities to obtain the final forecast.
The same approach can be applied to any additive or multiplicative feature of a time series. It is can also be generalised to a typical strategy of refining forecasts:
- Find some way to forecast a time series that is better than a memoryless, stationary random process.
- Take the difference between predicted mean and observation (a.k.a. forecasting error, residual) as a new time series.
- Go to step 1.