Remove the trees, you lose the leaves but you instead get landslips as the roots that held the embankments together die. Perhaps replanting with conifers would make sense.
Most tree species are robust and the roots do not die when the tree is pruned. Cutting trees back close to the ground is often described as rejuvenation pruning as the extant root system pushes out multiple new shoots from the rim of the severed trunk. This results in a fan of multiple trunks and is known as coppicing. The technique was harnessed by charcoal makers for many years as a method of generating easily accessible raw material. Successful rejuvenation does depend on the health of the tree. Recent years have seen the arrival of pests and diseases that have struck down Elm, Ash. Oak and now Horse Chestnut Trees. Such diseased trees represent a hazard to the safe operating of the railway and need to be removed. Anything that might compromise overhead electrification is regularly scrutinised. Hence current works between Bolton and Manchester.
Tree preservation Orders are not a significant problem. All Councils who issue these orders have arboriculture officers whose agreement is required to all works on preserved trees. In cases where trees are encroaching on houses or industrial/railway property, such consent is not unreasonably withheld. They also are concerned about who will being doing the work lest the tree is left susceptible to infection by air borne fungal spores,etc. IIRC not only do Network Rail have their own Orange Army arboriculture specialists and but also they employ specialist contractors who know their stuff.
The major problem about tree clearance tends to come from neighbours whose immediate response is to rattle the bars of their local MP who keen to be seen to be doing something immediately contact the local press. Often complaints are couched in terms of environmental vandalism, destruction of nature corridors etc. These arguments tend to be flawed because nature is resilient and as indicated above tree rejuvenation quickly follows. Major projects often schedule preparatory works to take place outside of the bird nesting season to minimise damage to the bird population. The scale and limited nature of lineside tree pruning means that small mammals can move along the corridor. Wild life also takes advantage of cleared areas. Insect life is often given a boost as contractors try to reduce the amount of material they must remove from the site by using chippers which provide a supply of biodegradable weed suppressant, plant food and habitat for creepy crawlies who then form a food supply for the next level in the food chain.
Although many people couch their complaints in environmental terms; they are, however, mainly are concerned about the personal impact of such works in terms of loss of privacy with train passengers with their noses stuck to carriage windows staring at them as trains hurtle past. Alternatively, they complain about the noise or vibration which do not change materially with the removal of the tree.
The view from their house may be changed by the removal of trees but residents such as appear in the earlier images seen above seem to forget that you cannot own the view from your house unless you own the land containing the view. I cannot tell my neighbour that he cannot have a trampoline in his garden (although aged seventy-six, I cannot foresee that as being a priority for him). Most railways in this country were built in the mid-19th century and few if any of the original residents are around to complain about the over the fence activity. The inference is that current residents must have bought their property whilst there was a railway running at the back of it. If they knew it was there when they bought the house, then they knew or ought to have known that from time to time such a railway would need maintaining including the embankments.
There is an old story, which may be apocryphal about a passenger on an American rail-road who was bitten by bed bugs and who wrote in to complain. He received in return a fulsome apology saying that this was a regrettable but isolated incident and hoped that he would travel with them again. Having been mollified by the letter, his mood changed when a manuscript note dropped out of the envelope an instruction from the manager to the typist which read: Send the bed-bugs letter. I do hope that lurking on the Network Rail word processing system that there is the equivalent Tree Hugs letter.