$~$\\mathcal L(H \\mid e) < 0.05$~$ also doesn't mean "$~$H$~$ is less than 5% likely", and the student still needs to learn to keep "probability of $~$e$~$ given $~$H$~$" and "probability of $~$H$~$ given $~$e$~$" distinctly separate in their heads\. However, likelihood functions do have a simpler interpretation: $~$\\mathcal L(H \\mid e)$~$ is simply the probability of the actual data $~$e$~$ occurring if $~$H$~$ were in fact true\. No need to talk about experimental design, no need to choose a summary statistic, no need to talk about what "would have happened\." Just look at how much probability each hypothesis assigned to the actual data; that's your likelihood function\.
Have I gone mad, or do you mean "L(H|e) is simply the probability of H given that the the actual data e occurred"?