Locking nuts do make stringing easier as you don't need to wrap the string round the post, just pull it through the hole til there's very little slack, lock it in place & tune it to pitch in a few turns.
However, the majority of tuning problems come from the nut. The pencil-lead in the nut slot trick Dakine referred to is easy & cheap to try (for best results use a really soft lead pencil from an arts/crafts shop) - the graphite reduces friction between the string & the nut. You can also get dedicated lubircants such as Big Bends Nut Sauce.
I guess any repairman/luthier can replace a nut, but I'd prefer to have someone I can trust work on my guitars - the repairman I use charges around £40
I've had very few guitars where the machineheads themselves have been responsible for tuning problems - sods law, one of them is my Nailbomb Les Paul - one of my favourite guitars :roll: but beware that you may need to ream the tuner holes out and/or drill guideholes to fit different tuners to your guitar....