You have a linear ODE with explicit coefficients: the solutions can actually be written down explicitly via variation of constants. To summarize the result, let $g_{a,b}(r)$ be given by

$$ g_{a,b}(r) = \frac{a}{r^{\ell+1}} + \frac{b}{r^2} $$

you find that

$$ r^2 g_{a,b}'' + 2r g_{a,b}' - \ell(\ell+1) g_{a,b} = -\frac{(\ell+2)(\ell-1) b}{r^2} $$

As

$$ g_{a,b}(1) = a + b , \qquad g'_{a,b}(1) = -(\ell+1) a - 2b $$

for $g_{a,b}$ to solve the equation you indicated requires $(\ell+2)\ell-1)b = g_{a,b}(1) + g'_{a,b}(1)$. So we need (when $\ell > 0$)

$$ - \ell a - b = (\ell+2)(\ell - 1)b \implies a = \frac{1 - \ell - \ell^2}{\ell} b $$

which yields

$$ \alpha = \frac{1-\ell^2}{\ell} b, \qquad \beta = - \frac{b}{\ell}(1 + 2\ell - 2\ell^2 - \ell^3) $$

so the Dirichlet to Neumann map has norm exactly

$$ \frac{\ell^3 - 2\ell^2 + 2\ell + 1}{\ell^2 - 1} = O(\ell) = O(\sqrt{\ell(\ell+1)})$$

You have a linear ODE with explicit coefficients: the solutions can actually be written down explicitly via variation of constants. To summarize the result¹ when $\ell > 1$, let $g_{a,b}(r)$ be given by

$$ g_{a,b}(r) = \frac{a}{r^{\ell+1}} + \frac{b}{r^2} $$

you find that

$$ r^2 g_{a,b}'' + 2r g_{a,b}' - \ell(\ell+1) g_{a,b} = -\frac{(\ell+2)(\ell-1) b}{r^2} $$

As

$$ g_{a,b}(1) = a + b , \qquad g'_{a,b}(1) = -(\ell+1) a - 2b $$

for $g_{a,b}$ to solve the equation you indicated requires $(\ell+2)\ell-1)b = g_{a,b}(1) + g'_{a,b}(1)$. So we need (when $\ell > 1$)

$$ - \ell a - b = (\ell+2)(\ell - 1)b \implies a = \frac{1 - \ell - \ell^2}{\ell} b $$

which yields

$$ \alpha = \frac{1-\ell^2}{\ell} b, \qquad \beta = - \frac{b}{\ell}(1 + 2\ell - 2\ell^2 - \ell^3) $$

so the Dirichlet to Neumann map has norm exactly

$$ \frac{\ell^3 - 2\ell^2 + 2\ell + 1}{\ell^2 - 1} = O(\ell) = O(\sqrt{\ell(\ell+1)})$$

¹ The cases where $\ell = 0$ and $1$ requires more care.

When $\ell = 0$, the equation you wrote is only solvable when $b = 0$, and the Dirichlet-to-Neumann map has norm 1.

When $\ell = 1$, variation of constants gives $$ g_{a,b} = \frac{1}{r^2}(a + b\ln(r)) $$ for which $$ r^2 g_{a,b}'' + 2r g'_{a,b} - 2 g_{a,b} = -\frac{3b}{r^2} $$ which then requires $a = -2 b$, and hence $\alpha = -2b$ and $\beta = 5b$, and the D2N map has norm $5/2$.