Part of the problem was that the line and frame oscillators were made from discrete components, that is plain old resistors, transistors(or valves in older sets), capacitors.These components especially as they age are affected, their design values change and the oscillation frequency changes.
Old valve tv sets were worse because there was a lot of heat generated which ages components and the valves themselves degrade over time.
The idea was that they provided the frequencies needed for those functions but were also given a correcting ‘kick’ by components of the recieved signal.
Think of it like a weight on a long spring, you pull the weight downwards and let go, it bobs up and down at a particular speed, but it won’t keep going for ever so you(the oscillator) give it a little push at the end of each bounce.
Now imagine that over time the spring suffers a little bit of metal fatigue so that it bounces at a differant speed, you keep putting some energy into the system and it keeps it correct.You get your timing for this from the recieved signal.Eventually the springs properties change so much that you cannot keep up the same speed, so you adjust the length of the spring.
If the input of energy is not great enough to correct the the oscillation it will just free run at some other speed.Hence the rolling and tearing of the image.
That control is adjusting the circuit to compensate for changes in circuit conditions.
Modern tv sets have oscillators that use small pieces of quartz crystal in them. These are incredibly accurate and stable, their properties to all intents and purposes just do not change.
The local oscillator is not responsible for line and frame control.Not in a direct sense anyway.
Your aerial picks up every tv signal at once and passes it down the cable to the tv tuner.
Each station has a differant frequency and when the whole bunch is recieved it is mixed with another signal, the local oscillator, to produce a harmonic signal which is further processed.
This harmonic is fixed in frequency, what you do is to change the frequency of the local oscillator every time you change the station.This produces what is called an I.F stage, intermediate frequency.This is what allows the tv to separate one station from another.
This process is called heterodyning, or super heterodyning as there is usually more than one stage of frequency mixing.