I think the arguments are far too polarized in favour and against cross channel checks.
In good conditions the vast majority of new PPLs will cope well. In poor conditions many will struggle. In spite of the training flying on instruments for 30 minutes or more can be disconcerting. Moreover, it is not just the cross channel aspect but how many pilots are actually shown how to file a flight plan or what to expect when dealing with French air traffic. This means that the more confident new PPL will cope albeit find the experience a new challenge. The less confident will find the experience disconcerting, and would have been happy to have had along an instructor or more experienced pilot the first time. Of course this is where a good club and good instructors will know their students and will suggest with some x students they might benefit from being accompanied. In short there is no right or wrong answer.
A point that hasn’t been made is that when accompanied whatever the debate about how to log the flight what should have been clear before the flight even started is the relationship between the crew. Who is in command? Is the pilot under training? At the very least the instructor should make this absolutely clear.
An earlier post passed comment that it is not rocket science and all will be well so long as you have life jackets. I have to disagree. Do not kid yourself life jackets are of very little value. For most of the year in the very unlikely event you come down on the sea in the channel with only a life jacket there is a very good chance you will die of hypothermia before you are rescued. The risk assessment is yours of the likelihood of an engine failure but if you don’t like the answer life jackets are not the solution - invest in a life raft!