What happens should the recording company, complying with the regulations go bust overnight?
Good point. However, just because the company is bust does not mean that the records that they are legally required to retain simply disappear.
Same applies to the accounts, director reports, annual returns, aircraft records, crew records, Health and Safety etc etc etc.
There is provision for people who are in the position of having lost their logbook or had it destroyed to make a Legal Declaration as to their hours. Every 5 years when I renew my licence the Authority takes a photocopy of the last 3 pages of my logbook so worst case scenario is having to remember the last 5 years.............which if you have the ATPL and all the experience requirements for command, instructor, examminer completed many moons ago is not much of a problem anyway.
For all the important issues in commercial flying there is a closed loop containing the TRTO, The AOC Holder and the Authority i.e. you complete your course and the TRTO pass the required paperwork to the AOC holder who pass it to the Authority who check their records and if all is OK post you new page(s) for your licence i.e. you don't have to produce your logbook.
Personally, I have always kept my own logbook because experience has shown that some companies do not record the times correctly i.e. they record times in accordance with National procedures rather than JAR-FCL.
Finally, just because it is legal for the company to act as logbook recorder does not guarantee that it is a good idea!!