Xenia, Xenia, Xenia.
It is NOT as easy as that to move when you live in a council or HA property. You have to have a LOCAL CONNECTION to the area to be able to have a council or HA property there. A local connection is only : Mother, Father, Brother, Sister living in the area.
Without a local connection, if you turn up at a council from elsewhere, without a family connection in that area, and you're not fleeing DV, you can't get a council or HA house.
If you are currently in a council or HA house, you can't just go private rent somewhere else because you need to (mostly) be in employment to privately rent, or have a guarantor. Which a lot of people won't have. You also need to find up to 6 months rent up front if you are unemployed.
You can't undertake cash in hand leaf letting work, or any other cash in hand work, while you are claiming benefits - that is benefit fraud unless declared, and if declared, you will be making yourself possibly not eligible to claim JSA if they deem that you are unavailable for FT work as you are working too many hours. If you declare it, all that will happen is they will take the cash off your benefits, except the first £20 a week of any earnings.
An Internet connection isn't the only thing needed to do proof reading online - a good standard of English is also required. And they still have the issue of it possibly stopping their benefits - which also includes their Housing Benefit which secures the roof over their family's heads.
Also, if you take a short-term, temporary job, it can stop your housing benefit for the duration of the job, while they 'reassess' your claim to see what portion of your rent will then be covered. It is currently taking 24 weeks for a Housing Benefit reassessment in my area. If you start a temporary job, that's one reassessment. When that job stops, another reassessment is needed. But they still haven't reassessed the FIRST time yet.
People who rely on top-up benefits like Housing Benefit and Tax Credits just CAN'T take temporary jobs, as they will lose their home. Their wages won't even cover the rent of a 2-bed council/HA flat here. Let alone such fripperies as food, electric, gas and water.
You REALLY have no idea what you are talking about, do you?