When examining global error bounds for Euler method, can I rescale the domain limits?

When examining global error bounds for Euler method, can I rescale the
domain limits?

I'm looking at provable global error bounds of the Euler method for the
first time and I was surprised to find that the bound grows exponentially
in the amount of time (the domain size) propagated i.e.
$\delta_{err}\le A \;dt\; (e^{L (t_1-t_0)}-1)$
With $A$ a function of problem dependent bounds (e.g. the Lipschitz
constant, $L$). At first, I thought this was extremely bad due to the
exponential dependence on the domain size but then it occurred to me that
I could rescale the time units such that $(t_1-t_0)=1$ but this seems like
it might be problematic.
Going beyond Euler method to Runge-Kutta etc. methods doesn't help if the
error bound grows exponentially with the domain size. As ODE is a well
established field, and most references don't explicitly mention the
exponential growth with the domain size I think it must be unimportant but
I cannot see how (other than rescaling). Hence my question is:
Is there a way to be okay with this exponential dependence, either by
rescaling such that $t_1-t_0=1$ or by any other means?