1. Here is an elementary and constructive proof from a hermitian point of view.

I will reproduce it only in the case of curves and blow-up equal to the identity, the general case being just more complicated to write.

Let $C$ be a compact curve and $A\to C$ a positive line bundle. If the uniformity is not true, then there exists a sequence $(p_k)\subset C$ such that $$ A^{\otimes k}\otimes\mathcal O_C(-p_k) $$ is not positive. Up to subsequence we can suppose that $p_k\to p\in C$. Let $k_0$ be an integer such that $$ A^{\otimes k_0}\otimes\mathcal O_C(-p) $$ is positive when $\mathcal O_C(-p)$ is endowed with the following metric:

Take a trivialization of $\mathcal O_C(-p)$ consisting in a coordinate chart $U$ centered in $p$ and in the open set $V_p=C\setminus\{p\}$ complementary of the point $p$. Put the trivial metrics $h_{U}$ and $h_{V_p}$ on this two trivialization and glue them together with the very simple partition of unity given by a smooth function $\vartheta\colon C\to\mathbb[0,1]$ which has compact support contained in $U$ and is identically equal to $1$ in an open neighborhood $\Omega$ of $p$; the metric on $\mathcal O_C(-p)$ is thus given by $h_{\mathcal O_C(-p)}=\vartheta\,h_U+(1-\vartheta)\,h_{V_p}$.

Then, for every $q$ in the aforesaid open neighborhood we can put the metric $h_{\mathcal O_C(-q)}$ on $\mathcal O_C(-q)$ given by $\vartheta\,h_U+(1-\vartheta)\,h_{V_q}$ where $h_{V_q}$ is the constant metric on the trivialization $V_q=C\setminus\{q\}$ of $\mathcal O_C(-q)$.

These two metrics are in fact equal Therefore, the two curvatures coincide and $$ A^{\otimes k_0}\otimes\mathcal O_C(-q) $$ is positive, contradiction.

2. If you prefer a less constructive and more global approach, let $p$ and $q$ two distinct points on a compact curve $C$, the $\mathcal O_C(p-q)$ has zero first Chern class, and thus this line bundle admits a metric $h$ of identically zero curvature (this is just the $\partial\bar\partial$-lemma). Now put any metric on $h_q$ on $\mathcal O_C(q)$. Then $h_p=h\otimes h_q$ is a metric on $\mathcal O_C(p)$.

If $A^{\otimes k}\otimes\mathcal O_C(-p)$ is positive, that is if $$ k\,i\,\Theta(A)+i\,\Theta(\mathcal O_C(-p),h_p^{-1}) $$ is a positive definite $(1,1)$-form, then $$ \begin{multline} k\,i\,\Theta(A)+i\,\Theta(\mathcal O_C(-q),h_q^{-1}) \\ = k\,i\,\Theta(A)-i\,\Theta(\mathcal O_C(-p),h_p^{-1})+i\,\Theta(\mathcal O_C(-p),h_p^{-1})+i\,\Theta(\mathcal O_C(-q),h_q^{-1}) \\ = k\,i\,\Theta(A)+i\,\Theta(\mathcal O_C(p),h_p)+i\,\Theta(\mathcal O_C(-p),h_p^{-1})+i\,\Theta(\mathcal O_C(-q),h_q^{-1}) \\ = k\,i\,\Theta(A)+i\,\Theta(\mathcal O_C(p-q),\underbrace{h_p\otimes h_q^{-1}}_{=h})+i\,\Theta(\mathcal O_C(-p),h_p^{-1}) \\ = k\,i\,\Theta(A)+i\,\Theta(\mathcal O_C(-p),h_p^{-1}) >0. \end{multline} $$

1. Here is an elementary and constructive proof from a hermitian point of view.

I will reproduce it only in the case of curves and blow-up equal to the identity, the general case being just more complicated to write.

Let $C$ be a compact curve and $A\to C$ a positive line bundle. If the uniformity is not true, then there exists a sequence $(p_k)\subset C$ such that $$ A^{\otimes k}\otimes\mathcal O_C(-p_k) $$ is not positive. Up to subsequence we can suppose that $p_k\to p\in C$. Let $k_0$ be an integer such that $$ A^{\otimes k_0}\otimes\mathcal O_C(-p) $$ is positive when $\mathcal O_C(-p)$ is endowed with the following metric:

Take a trivialization of $\mathcal O_C(-p)$ consisting in a coordinate chart $U$ centered in $p$ and in the open set $V_p=C\setminus\{p\}$ complementary of the point $p$. Put the trivial metrics $h_{U}$ and $h_{V_p}$ on this two trivialization and glue them together with the very simple partition of unity given by a smooth function $\vartheta\colon C\to\mathbb[0,1]$ which has compact support contained in $U$ and is identically equal to $1$ in an open neighborhood $\Omega$ of $p$; the metric on $\mathcal O_C(-p)$ is thus given by $h_{\mathcal O_C(-p)}=\vartheta\,h_U+(1-\vartheta)\,h_{V_p}$.

Then, for every $q$ in the aforesaid open neighborhood we can put the metric $h_{\mathcal O_C(-q)}$ on $\mathcal O_C(-q)$ given by $\vartheta\,h_U+(1-\vartheta)\,h_{V_q}$ where $h_{V_q}$ is the constant metric on the trivialization $V_q=C\setminus\{q\}$ of $\mathcal O_C(-q)$.

These two metrics are in fact equal Therefore, the two curvatures coincide and $$ A^{\otimes k_0}\otimes\mathcal O_C(-q) $$ is positive, contradiction.

2. If you prefer a less constructive and more global approach, here it is:

Let $p$ and $q$ two distinct points on a compact curve $C$, the $\mathcal O_C(p-q)$ has zero first Chern class, and thus this line bundle admits a metric $h$ of identically zero curvature $i\,\Theta(\mathcal O_C(p-q),h)$ (this is just the $\partial\bar\partial$-lemma). Now put any smooth hermitian metric on $h_q$ on $\mathcal O_C(q)$. Then $h_p=h\otimes h_q$ is a metric on $\mathcal O_C(p)$.

Therefore $$ \begin{aligned} i\,\Theta(\mathcal O_C(p),h_p)&=i\,\Theta(\mathcal O_C(p),h_p)-i\,\Theta(\mathcal O_C(q),h_q)+i\,\Theta(\mathcal O_C(q),h_q) \\ &=i\,\Theta(\mathcal O_C(p-q),\underbrace{h_p\otimes h_q^{-1}}_{=h})+i\,\Theta(\mathcal O_C(q),h_q)\\ &=i\,\Theta(\mathcal O_C(q),h_q), \end{aligned} $$ so that the same constant $k_0$ can be taken for all prime divisor $p$ in order to have that $$ A^{\otimes k}\otimes\mathcal O_C(-p) $$ has positive definite Chern curvature.