Let $X$ be a connected smooth scheme over a field and let $VB(X)$ be the exact category of vector bundles (i.e. locally free $\mathcal{O}_X$-module whose rank is finite at every point) over $X$.

Let the abelian group $\tilde K_0(X)$ denote the kernel of the rank morphism from the zeroth algebraic K-theory group $K_0(X)$ to $\mathbb{Z}$.

Let $Vec_k(X)$ be the set of isomorphism classes $[A]$ of rank $k$ vector bundles $A$ over $X$. One gets maps $i_k:Vec_k(X)\to Vec_{k+1}(X)$ defined by $[A]\mapsto [A\oplus \mathbb{A}^1_X]$, i.e. by adding a trivial bundle. The directed colimit $cVec(X)=\operatorname{colim}_k Vec_k(X)$ of this system has the structure of a commutative monoid by $[A]+[B]=[A\oplus B]$.

Is there an isomorphism $cVec(X)\cong\tilde K_0(X)$ of monoids?

This holds for topological K-theory over a connected CW-complex. I am a little unconvinced that this isomorphism holds since $VB(X)$ is not split-exact. However, I cannot find an argument.