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As apparently seems par for this book, the theorem statement you're paraphrasing involves vaguely defined or completely undefined terms. For example, it talks about "*admissible* (smooth) paths" but this is the first use of the term "admissible" in the text.^[The notation $f'_q$ is bizarre. The authors like to use $'$ to indicate differentiation with respect to whatever they feel like at the moment, but this is clearly a problem for multiparameter functions. So instead of explicitly writing $\frac{\partial f}{\partial q}$ or following other authors and writing $f\_q$, they use $f'\_q$ presumably to (help) distinguish subscripts that are indexes from subscripts that mean differentiation. Unfortunately, they don't seem to articulate this convention anywhere, they also use $'$ for just distinct but related variables, and they (presumably) typo the definition of "Lagrangian derivative" as $(L'\_{{\dot q}\_i})^\cdot - L\_{q\_i}$ where $(L'\_{{\dot q}\_i})^\cdot - L'\_{q\_i}$ was presumably intended.]

Anyway, the situation you describe violates the assumptions that the covectors are linearly independent. Even in the case where you only have one, the set $\\{0\\}$ is a linearly dependent set.

More practically, constraints of the sort you describe, i.e. that are functions of the (generalized) positions (and possibly time) alone, are called [holonomic](https://en.wikipedia.org/wiki/Holonomic_constraints). If you think about, what it means to say that $f(\mathbf q, t) = 0$, an acceptable constraint of the form the authors describe arises. The constraint $f(\mathbf q, t) = 0$ states that at each point $\mathbf q$, $f$ is a conserved quantity. This then gives rise to a [continuity equation](https://en.wikipedia.org/wiki/Continuity_equation) expressing this conservation, namely $\dot{\mathbf q} \cdot \nabla_\mathbf q f + \frac{\partial f}{\partial t} = 0$. That is, the amount of $f$ flowing into a point $\mathbf q$ has to exactly match the amount that $f$ decreases with time. We can then define $g(\mathbf q, \dot{\mathbf q}, t) = \dot{\mathbf q} \cdot \nabla_\mathbf q f + \frac{\partial f}{\partial t}$ and use this as a constraint. It will typically give rise to a non-zero gradient with respect to $\dot{\mathbf q}$.

For the case where $f$ is not time-varying. The continuity equation states that the only allowed tangent vectors are those normal to the gradient, i.e. those in the directions of constant $f$. In this case, the $\mathbf v$ such that $g(\mathbf q, \mathbf v) = 0$ pick out a (hyper-)plane through the origin which, if translated to $\mathbf q$, would be tangent to the (hyper-)surface $f(\mathbf q) = 0$ at $\mathbf q$. This is exactly the tangent space we'd want in this situation. The gradient of $g(\mathbf q, \mathbf v)$ with respect to $\mathbf v$ is just $\nabla_\mathbf q f$ again. The virtual velocity vectors are again just the vectors of this tangent (hyper-)plane.

If $f$ is time-varying, then at a fixed point and time, $\mathbf q$ and $t$, the $\mathbf v$ such that $g(\mathbf q, \mathbf v, t) = 0$ will again pick out a (hyper-)plane, but this time it will not be a (hyper-)plane through the origin (unless $\frac{\partial f}{\partial t} = 0$). However, $\nabla_\mathbf v g(\mathbf q, \mathbf v, t)$ will still be $\nabla_\mathbf q f$ and thus the virtual velocities will be the same as if $f$ wasn't time-varying but instead was frozen at its current shape.