Well, I'd like to give my personal idea:
Since we have $$\dot{x}(t)=\sum_{i=1}^m[({\partial \phi\over \partial q _ k})\circ g](t)\cdot\dot{q_i}(t)+({\partial \phi\over \partial {t}})\circ g(t), \forall t\in \mathbb{R},$$ Then if we define the following maps: $$ \mu:\mathbb{R}^{2m+1}\to \mathbb{R},(y_1,y_2,\cdots,y_{2m+1})\mapsto \sum_{i=1}^my_{m+i}\cdot y_i\ +y_{2m+1},and$$$$\Pi:\mathbb{R}\to \mathbb{R}^{2m+1}, t\mapsto\left(\dot{q_1}(t),\cdots,\dot{q_m}(t),[({\partial \phi\over \partial q _ 1})\circ g](t),\cdots,[({\partial \phi\over \partial q _ m})\circ g](t),[({\partial \phi\over \partial {t}})\circ g](t)\right) , $$ then we have $$\dot{x}(t)=(\mu\circ\Pi)(t),\forall t\in \mathbb{R}$$ thus $\dot{x}=\mu\circ\Pi.$ And we can see $$ {\partial \mu\over \partial y_ k}(z_1,z_2,\cdots,z_{2m+1})=z_{m+k},\forall 1\leqslant k \leqslant m,$$ thus$$\left({\partial \mu\over \partial y_ k}\circ\Pi\right)(t)=[({\partial \phi\over \partial q _ k})\circ g](t),\forall 1\leqslant k\leqslant m.$$ So when we say $${\partial\dot{ x}\over \partial \dot{q _ k}}={\partial x\over\partial q_ k},$$our actual meaning is that $$\left({\partial \mu\over \partial y_ k}\circ\Pi\right)=[({\partial \phi\over \partial q _ k})\circ g].$$ But I don't this explanation is convincing, and I do think that the symbol $${\partial\dot{x}\over \partial \dot{q _ k}}$$is actually very misleading.

Well, I'd like to give my personal idea:
Since we have $$\dot{x}(t)=\sum_{i=1}^m[({\partial \phi\over \partial q _ k})\circ g](t)\cdot\dot{q_i}(t)+({\partial \phi\over \partial {t}})\circ g(t), \forall t\in \mathbb{R},$$ Then if we define the following maps: $$ \mu:\mathbb{R}^{2m+1}\to \mathbb{R},(y_1,y_2,\cdots,y_{2m+1})\mapsto \sum_{i=1}^my_{m+i}\cdot y_i\ +y_{2m+1},and$$$$\Pi:\mathbb{R}\to \mathbb{R}^{2m+1}, t\mapsto\left(\dot{q_1}(t),\cdots,\dot{q_m}(t),[({\partial \phi\over \partial q _ 1})\circ g](t),\cdots,[({\partial \phi\over \partial q _ m})\circ g](t),[({\partial \phi\over \partial {t}})\circ g](t)\right) , $$ then we have $$\dot{x}(t)=(\mu\circ\Pi)(t),\forall t\in \mathbb{R}$$ thus $\dot{x}=\mu\circ\Pi.$ And we can see $$ {\partial \mu\over \partial y_ k}(z_1,z_2,\cdots,z_{2m+1})=z_{m+k},\forall 1\leqslant k \leqslant m,$$ thus$$\left({\partial \mu\over \partial y_ k}\circ\Pi\right)(t)=[({\partial \phi\over \partial q _ k})\circ g](t),\forall 1\leqslant k\leqslant m.$$ So when we say $${\partial\dot{ x}\over \partial \dot{q _ k}}={\partial x\over\partial q_ k},$$our actual meaning is that $$\left({\partial \mu\over \partial y_ k}\circ\Pi\right)=[({\partial \phi\over \partial q _ k})\circ g].$$ We also have $\dot{x}=\mu\circ\Pi ,and \ x=\phi\circ g, $ which is interesting.

But I don't this explanation is convincing, and I do think that the symbol $${\partial\dot{x}\over \partial \dot{q _ k}}$$is actually very misleading.