Like the poster "Livebox", I had one from about 2006 to 2012, without problem.
Incidentally, Switch 2 simply stops the varying level if the blue LED inside the Livebox - no idea why that LED was fitted, with the complications of the switch etc. Did not have any real function that I am aware of.
Regarding your own problems, Switch 1 is normally only for use when adding a (new) WiFi device, raising the possibility that it may not have been properly included in the first place.
How many and what type of WiFi devices have you connecting through the Livebox?
Have you tried working through all of those devices when you encounter such a loss, WITHOUT pressing Switch 1 - ie is it total loss of the Livebox WiFi function; and ONLY that?
(If you have a Laptop/Notebook/Netbook, download the WiFi monitoring software from Metageek, to see whether the Livebox WiFi is functioning when you have a disconnection avoiding Switch 1 - also try first when everything is OK, so that you know what you are looking for when a subsequent disconnect occurs. Very easy to use!)
Does any Ethernet wired-device lose access to the Web simultaneously?
Although at present we don't know the cause, you should also try the Phone Quiet Test, by unplugging everything at the Phone Master Socket, plugging in the simplest (probably oldest) phone you have, in that Master Socket.
Choose Option 2 Quiet Test and listen carefully for noises, signals etc - there SHOULD BE NONE.
Note that some lines when connected via ISPs such as SKY, can not acess this test - but at first sight, yous should.
For the following tests, you should Highlight/Edit-Copy/Edit-Paste the results in to a suitable Document such as in WORD or Notepad.
Makes life a lot easier.
With your phone system, computers etc returned to normal and working, try the BRAS Test at-
and note the results, particularly the Line Distance and the Distribution Cabinet Number - these can be very useful.
For comparison, use Street View or Google Earth, to measure the straight-line/crow's path to your Exchange; and also use the Route Finder part to get the likely road distance, for comparison with the BRAS lLne Length.
With a PC/Laptop connected by Ethernet cable to the Livebox, access its Admin function.
Find the page with information such as Line Attenuation etc, both Downstream and Upstream. making a note of them.
Post all of those results back on this Thread.
It may seem lengthy; but most if not all may be required to solve you problem; and doesn't take long.
For example, if it is a faulty line, the Brightbox although a significant improvement compared to the Livebox, would also lose connection - and any other Router/Modem from any other ISP.