If all variables of a system was known, would stochastic modelling be redundant? Statistical parametric methods such as regression can be used to predict a particular dependent variable using a pre-defined set of covariates. For example, a linear regression model used to predict merchandise sales.
If all covariates that cause true value of the continuous dependent variable to change was known, more specifically if an event could be deterministically modelled, then would using parametric modelling techniques to represent such event be unnecessary since a deterministic model would always yield an accurate prediction for any observation?
Thus, in predictive modelling, should more time be devoted in collating and gathering as much relevant features to increase the accuracy of a model rather than trying to optimise a model based on a given pre-defined list of independent random variables?
 A: It's a philosophical one. Are you ever able to observe all the causes? Say that you are modeling sales in a local grocery store. There are some obvious factors that would influence sales, but they do not explain all the variability. Say a severe snowstorm hits, so people are not able to commute to the store, and sales drop. There is an economic crisis, or pandemic, that makes people's behavior change. Maybe there was opened another store nearby and they have attractive prices this week to attract customers. Or maybe they are renovating the main road leading to the store, so your customers are preferring other stores to spend less time in traffic jams. To model sales in the local grocery store, would you collect all the data on all the economic, social, weather, political, medical, etc conditions related to all the possible customers of your store? I doubt and most likely you won't be able to collect all the data, even if you tried. You need statistical models because it is impossible to control for all the factors that would influence the phenomenon you are studying, and there is also a measurement error to account for.
A: Typically we assume we see $n$ i.i.d samples of the form $(X_1, \dots, X_p, Y)$ from some joint distribution. Suppose we think we're observing enough meaningful covariates $X_1 ,\dots, X_p$ that $Y$ is truly a deterministic function of them, so $Y = f^{*}(X_1, \dots, X_p)$ for some function $f^{*}$. This is distinct from the more typical additive model assumption that $Y = f^{*}(X_1, \dots, X_p) + \epsilon$ where $\epsilon$ is some independent noise.
Observing the data we have, we want to pick a function $f$ in some class of functions $\mathcal{F}$ such that $f(X_1 , \dots, X_p) \approx Y$ for new (not just training) samples. If again $\mathcal{F}$ is a parametric class, this is exactly a parametric prediction problem. The key difference is the randomness now solely comes from whatever training sample you drew, as the $f$ you choose will depend on this sample. Overfitting will not be a result of overfitting to noise, but fitting a function $f$ that matches $f^{*}$ on the training data but does not match $f^{*}$ for new samples.
If you want to completely remove randomness from the problem, you could imagine there's a fixed set of potential inputs and outputs $\{(x^{(i)}_1, \dots, x^{(i)}_p, y^{(i)}) \}_{i=1}^N$ (we could generalize to an infinite collection) that you're interested in and you want to find a function $f$ such that $f(x_1^{(i)}, \dots, x_{p}^{(i)}) = y^{(i)}$ for all $i$. The caveat is you only get to see $y^{(i)}$ for a subset of the fixed points. Your goal would likely to be to find a function $f$ that satisfies $f(x_1^{(i)}, \dots, x_{p}^{(i)}) = y^{(i)}$ for all the $i$ that you observe $y^{(i)}$ for, and is additionally constrained (e.g. it is smooth or $L$-lipschitz) in a way that leads you to think it'll have the desired property for the data points you didn't get to observe $y^{(i)}$ for as well.
To answer your final question, there are probably some situations, particularly areas where deep learning has been successful (like natural language process and computer vision), where we have "gathered" all the covariates needed to predict $Y$ deterministically. These are often deemed "high signal" tasks. But in many cases, it's often unknown to us what these covariates are/should be included in the model, and extensively searching for and including such covariates will likely result in our model learning a bunch of spurious, not useful correlations that won't generalize outside the training data (the reasoning for why is identical to $p$-hacking).
If the question is further what is the advantage of one approach (including randomness in your quantitative modeling) vs. the other (not including randomness in your quantitative modeling), making stochastic assumptions puts you in a statistical framework and allows you to do inference. You can make predictive intervals for new samples, confidence intervals for parameters, get guarantees about convergence from empirical risk minimization etc. Of course, these results only hold if the stochastic assumptions are correct, something which is often not true or hard to affirm (depending on the strength of the assumptions). Without the stochastic assumptions there's no notion of such things, but we don't have to worry about our stochastic assumptions holding.
A: Not necessarily.
I will even do you one better: If all variables in a system and all causal functions relating them, including error processes (if any), were fully known and parameterized predictive modeling might still be required.
Knowing direct effects does not in and of itself give you the integration and accumulation of indirect effect over time. Even an purely deterministic equations can provide challenges to prediction if there is any uncertainty about initial conditions or even minute error, such as precision or representation. This is a general insight known in many branches of science, from population ecology (May, 1973; Levins, 1974), to physics (Schultz, 1994; see also the three-body problem) and chaos theory (Sanderson, 2021).



References
Levins, R. (1974). The Qualitative Analysis of Partially Specified Systems. Annals of the New York Academy of Sciences, 231, 123–138.
May, R. M. (1973). Qualitative Stability in Model Ecosystems. Ecology, 54(3), 638–641.
Sanderson, G. (2021). Newton’s Fractal (which Newton knew nothing about) [YouTube video]. 3Blue1Brown. Run time: 26:05
Schulz, H. C., & Hilgenfeldt, S. (1994). Experimente zum Chaos [algorithmic translation to English]. Spektrum der Wissenschaft, 1, 72-81. [A nice interactive simulation of the systems described in this article]
