Did you read the link you posted? Because it covered the major mismanagement which killed four people and directly resulted in renationalisation of UK rail infrastructure.
It's also worth noting that, for the UK rail system, most passenger operators are actually owned by public operators from other countries (like DB with their Arriva brand, NS with Abellio, etc.).
Unfortunately accidents happen, there are much bigger accidents happened in complete public rails. But if you check the measures , overall safety has been increased a great deal, actually it is considered as the safest in Europe, And for most accounts there are improvements (Customer satisfaction is up, Level of traffic is up, Fare increases are lower than before, Average train age unchanged, Punctuality unchanged, Safety is increased, Efficiency is up)
).
If it would be a real privatization, chances that it would be better.
It's also worth noting that, for the UK rail system, most passenger operators are actually owned by public operators from other countries (like DB with their Arriva brand, NS with Abellio, etc.).