The first step is to get the lost paths reinstated in the Definitive Map. This is a paper exercise. I would have thought that the burden of proof issue was fairly straightforward given where the data on these lost ways are drawn from. The sources are documentary rather than anecdotal. Here is the old map on which the path is shown, here is the Definitive Map on which the path is missing. The only reason why a path should not be accepted onto the Definitive Map should be that it is no longer physically possible e.g. a housing development, industrial estate, reservoir or motorway in the way.
Once they have been accepted onto the Definitive Map the rest can follow in slower time and the paths that need to be, can be reinstated in order of those that offer greatest benefits first. How this is done and how it is funded can be argued about once the Definitive Map has been updated.
IMHO the addition of other lost ways that are not marked as FPs on the old maps and the creation of new through routes is a separate issue requiring a greater burden of proof.
If only it was that straight forward. As I understand it the maps may indicate a footpath was there but not necessarily a public right of way. A disclaimer to this effect appears on all OS maps.
More evidence is required and that is where the difficulty is. The whole subject can be incredibly complicated and is to my mind skewed very much in favour of the landowner. The best evidence is of course that of users but with most routes I have investigated there just aren't any alive now, as the paths have long since fallen out of use and were never included on the definitive map, often of course through the failures of the system.
Some years ago when first getting involved in lost ways I was told by a very well known Ramblers Footpath Officer who has got hundreds of ROW onto the definitive map that: "I warn you that your research may well end in nothing more than frustration". He was of course right but I learnt an enormous amount in going through records of all kinds, many of which I never knew existed.
Others herein may already know of this reference book but I have certainly found it useful in my research as it explains the process of researching lost rights of ways in great detail. Namely: "Rights of Way, Restoring the Record, by Sarah Bucks and Phil Wadey ISBN 978-0-9574036-1-1.
It's worth a read although I found it a bit repetitive in parts.