Rigorously speaking, yes, you are right if dealing with the Lagrangian function. Indeed E.-L. equations should be more properly written
$$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}^k}\right) - \frac{\partial L}{\partial {q}^k}= 0\:, \quad \frac{d q^k}{dt} = \dot{q}^k\quad k=1,\ldots, n\:.$$
In other words $\dot{q}^k$ becomes $\frac{d q^k}{dt}$ just along the solutions of the equations but, otherwise $\dot{q}^k$ and ${q}^k$ are independent variables.
This is because, geometrically speaking, $L$ is a map from the first jet bundle $j^1(S)$ where $T: S\to \mathbb R$ is the fiber bundle called spacetime of configurations, the basis $\mathbb R$ represents the axis of time whereas every fiber $T^{-1}(t)$ is the configuration space at time $t$. Natural local coordinates adapted to the fiber bundle structure are the standard coordinates $t, q^1,\ldots, q^n$.
The jet bundle $J^1(S)$ adds kinematic coordinates $\dot{q}^1,\ldots, \dot{q}^n$.
In this picture the identity, in local natural coordinates,
$$\frac{df(q(t),t)}{dt}=\sum_{k=1}^n\frac{\partial f}{\partial q}(q(t),t)\dot{q}^k(t) + \frac{\partial f}{\partial t}(q(t),t)$$
makes sense along the solutions of EL equations, but it does not without fixing a curve $q=q(t)$ (solution of EL equations or not) because the derivative in the left-hand side cannot be computed.
Nevertheless the formalism is constructed just to encourage this intuitive and effective interpretation since, after all it is quite harmless. One may define
something like $$\widetilde{\frac{df(q,t)}{dt}}=\sum_{k=1}^n\frac{\partial f}{\partial q}(q,t)\dot{q}^k + \frac{\partial f}{\partial t}(q,t)\:,$$
without fixing a section of $S$. As soon as a solution of EL is given, the notation becomes consistent with the standard one.
It is important to stress that if focusing on the action rather than the Lagrangian, in order to implement the variational principle, it is correct to always identify $\dot{q}^k$ with $\frac{dq^k}{dt}$, since the action is a functional over a space of curves and $\dot{q}^k=\frac{dq^k}{dt}$ is always assumed to be valid on each of theses curves no matter if they satisfy EL equations or not.
Best Answer
There is the physics SE feature of offering (in the column on the right hand side of the page) suggestions, under the heading 'Related'
In this particular case the column 'Related' offers a link to a 2011 question titled Invariance of Lagrange on addition of total time derivative of a function of coordinates and time
According to the answer to that question that invariance is in accordance with relativity of inertial motion.
Example:
The dynamics of juggling balls is independent of some constant relative velocity. You can be on the platform of a train station, or in a train moving at a constant velocity, the way the balls will move relative to you is the same. If you throw the ball straight up it will land back in your hand. This holds good for the members of the equivalence class of inertial coordinate systems.
The criterion of stationary action is that in order to find the true trajectory you find the point in variation space where the derivative of the Action $S$ (with respect to variation) is zero.
That is, in terms of stationary action the value of the Action $S$ is in itself not relevant. The relevant factor is the derivative of the Action $S$ with respect to variation.
As we know, taking a derivative is an operation that discards some information. That leaves room to reverse engineer ways in which one can modify the Lagrangian while still ending up at the same point in variation space.
For more on the concept of stationary action:
There is a 2021 answer by me about Hamilton's stationary action with a two stage derivation:
1 Derivation of the Work-Energy theorem from $F=ma$
2 Demonstration that in cases where the Work-Energy theorem holds good Hamilton's stationary action will hold good also
(Of course, the usual presentation starts with positing Hamilton's stationary action and next it is demonstrated that F=ma can be recovered from that. As pointed out by contributor Kevin Zhou (jan 16, 2020): "[...] in physics you can often run derivations in both directions.")