Asymptotic behavior of functions with not analytic expression
Is it safe?  No, of course, it's not safe.  There could always be ways that your experiments are misleading.  Maybe the worst case wasn't triggered by your particular workload or in your particular experiments.  Maybe there's a quadratic term with a small constant, which is negligible for the values of $n$ you tested in your experiments but becomes dominant for very large values of $n$.  You can't rule out those possibilities with experiments.

Are the experiments good enough?  That depends on the community, and for what purpose you want to use them, and what conclusions you want to support.  Can you conclude that the worst-case running time, over all possible inputs, is linear?  No.  Can you conclude that for workloads that are likely to arise in practice, the running time appears to be linear?  Perhaps.  Is this enough to conclude that your system or your method has merit or is a significant contribution over the state of the art?  Quite possibly.

There is no single answer about what is "acceptable".  Instead, I suggest you take a broader, more nuanced view.  Understand the strengths and limits of any method of analysis; understand what conclusions you can and can't support, using those methods; understand what claims you want to be able to make; understand the values of your audience nad the community you are targetting; and go from there.