I believe this line came within a whisker of being re-opened for passenger use in the early 90s, as Ivanhoe Line pt 2. A great shame it never came to pass when re-openings were cheaper and faster, as it would be rather useful.
Be rather useful for what? The whole line seems to be of little strategic importance.
The primarily important main lines in this country all radiate out of London. Let's call that "Nodule 1". The MML would be an example.
The secondarily important lines radiate out of either Manchester or Birmingham. The Leicester-Birmingham line would be an example.
The tertiarily important lines radiate out of other cities such as Bristol, Leicester, Norwich, etc.
The Leicester-Burton line is fourth category - as it doesn't link Leicester (where the junction is facing the wrong way), and Burton, Coalville, Ashby are hardly cities of any size.
It also suffers from mining subsidence, which means both speed restrictions and increased p-way costs, particularly if you wanted to upgrade it for passenger use.
Fix the junction at Leicester and it might be useful for commuting from Ashby, Coalville, into Leicester. But it's likely to generate congestion at Leicester London Road station. There may be a small market for peak trains from Coalville-London. And if you could maybe build a park and ride station off the M1 there, but that section of the M1 often resembles a car park in the morning anyway and there would be other challenges.
But otherwise, apart from the far eastern end between the MML and Bardon Hill which is used for freight, AFAICT it's pretty much useless.