Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.

The inclusion $A(x)^{op} \to O(x)^{op}$ is (homotopy) cofinal - by Quillen's theorem A, it suffices to show that for any open $U$, $A(x)_{/U}$ is weakly contractible. But it is a co-directed poset : for any $V,W \to U$, $V\cap W$ is an open and therefore contains some affine open $x\in O\subset V\cap W\subset U$; therefore it is weakly contractible.

It follows that, in the $(2,1)$-category of presentable $1$-categories, $colim_{U\in O(x)^{op}}QCoh(U) \simeq colim_{U\in A(x)^{op}} QCoh(U)$.

Now note that the latter diagram in fact lives in the category of $E_0$ presentable categories, that is, presentable categories with a chosen basepoint (namely $O_U$) that are equivalent to something of the form $(Mod_R,R)$ for some (commutative) ring $R$.

By a $(2,1)$-categorical analogue of theorem 4.8.5.11. in Lurie's Higher algebra (which says that $R\mapsto (Mod_R,R)$, as a functor from rings to $E_0$ presentable categories, is fully faithful and colimit preserving) (it's actually not an analogue, it follows strictly from this theorem by specializing to $Set$-modules in presentable $(\infty,1)$-categories), and using the fact that $A(x)$ is weakly contractible, so colimits over it in $E_0$-objects are computed underlying, it follows that $colim_{U\in A(x)^{op}} QCoh(U)\simeq colim_{U\in A(x)^{op}} Mod_{O_U(U)}$ is equivalent to modules over $colim_{U \in A(x)^{op}} O_U(U)= colim_{U\in A(x)^{op}} O_X(U)= O_{X,x}$.

Note that here the colimit is taken in ordinary associative rings, but it's a filtered colimit, so it can also be taken in commutative rings.

In conclusion, for any point $x$, $colim_{U\in O(x)^{op}} QCoh(U)$ is equivalent to modules over $O_{X,x}$.

If $x$ is a generic point, then $O(x)$ is the category of all opens, and so you get the corresponding statement for $colim_U QCoh(U)$.

Note that here I've used the following version of "$2-\mathrm{colim}$": homotopy colimits in the $(2,1)$-category (in the sense of "$(\infty,1)$-categories with $1$-truncated mapping spaces") of presentable $1$-categories. I'm assuming that this shouldn't matter too much

Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.

The inclusion $A(x)^{op} \to O(x)^{op}$ is (homotopy) cofinal - by Quillen's theorem A, it suffices to show that for any open $U$, $A(x)_{/U}$ is weakly contractible. But it is a co-directed poset : for any $V,W \to U$, $V\cap W$ is an open and therefore contains some affine open $x\in O\subset V\cap W\subset U$; therefore it is weakly contractible.

It follows that, in the $(2,1)$-category of presentable $1$-categories, $colim_{U\in O(x)^{op}}QCoh(U) \simeq colim_{U\in A(x)^{op}} QCoh(U)$.

Now note that the latter diagram in fact lives in the category of $E_0$ presentable categories, that is, presentable categories with a chosen basepoint (namely $O_U$) that are equivalent to something of the form $(Mod_R,R)$ for some (commutative) ring $R$.

By a $(2,1)$-categorical analogue of theorem 4.8.5.11. in Lurie's Higher algebra (which says that $R\mapsto (Mod_R,R)$, as a functor from rings to $E_0$ presentable categories, is fully faithful and colimit preserving) (it's actually not an analogue, it follows strictly from this theorem by specializing to $Set$-modules in presentable $(\infty,1)$-categories), and using the fact that $A(x)$ is weakly contractible, so colimits over it in $E_0$-objects are computed underlying, it follows that $colim_{U\in A(x)^{op}} QCoh(U)\simeq colim_{U\in A(x)^{op}} Mod_{O_U(U)}$ is equivalent to modules over $colim_{U \in A(x)^{op}} O_U(U)= colim_{U\in A(x)^{op}} O_X(U)= O_{X,x}$.

Note that here the colimit is taken in ordinary associative rings, but it's a filtered colimit, so it can also be taken in commutative rings.

In conclusion, for any point $x$, $colim_{U\in O(x)^{op}} QCoh(U)$ is equivalent to modules over $O_{X,x}$.

If $x$ is a generic point, then $O(x)$ is the category of all opens, and so you get the corresponding statement for $colim_U QCoh(U)$.

Note that here I've used the following version of "$2-\mathrm{colim}$": homotopy colimits in the $(2,1)$-category (in the sense of "$(\infty,1)$-categories with $1$-truncated mapping spaces") of presentable $1$-categories. I'm assuming that this shouldn't matter too much