Well, photosynthesis has been deployed some million years ago and works rather well. Stop deforestation, manage forest wisely and start using wood again. 1 m3 wood = 1 ton of CO2 sequestered.
And it isn't going to fix things in terms of sheer volume either. We've cut down forests and burnt fossil fuels. Replenishing forests only reversed the effects of the former and doesn't account for the carbon released via the latter.
This is patently false.
Yes each individual tree is only a temporary store (with a half-life somewhere around 100 years), but that's ok because the forest as a whole is continually generating trees, and sequester CO2 in the form of soil. Trees also create conditions that protect that soil from erosion and re-emission (protection from wind, rain, and evaporation).
The total anthropogenic CO2 emissions since the industrial revolution works out to only 30 tonnes of carbon per hectare, which is a few inches of topsoil or a few additional trees per hectare. It's easy to soak up that much carbon just in biomass on the surface.
Even in farmland the addition of selected "farmer's trees" (carub, honey locust, acacia, leucaena) can be used. These are selected species of tree that actually increase yield when planted into a field at 20-30/acre, despite casting some shade on the crops. These trees typically have deep roots that harvest minerals and recycle nutrients that would otherwise leach down to the aquifer, and their root system can coexist with crops right up to the butt of the tree. The nutrients are dropped as mulch, and they also make nutritious pods for animal feed.
Of course, you still need a lot of trees. The internets say that a rough rule of thumb for the amount of carbon in a tree is about a tonne. Therefore, given current carbon emissions of about 10^10 tonnes a year you'd need to plant more trees than that --- every year. (And then wait multiple decades. This isn't a short term solution.)
As you say, another reason to plant them is for local benefit. You don't mention that trees also regulate the local microclimate; they tend to make rain, for example. When Cyprus was deforested, it got a lot drier. I knew someone who was working to try and reforest a lot of it with the explicit goal of trying to fix some of the water supply issues; I don't know if they got anywhere. And, of course, they're just nice to be around.
Recent research show that there are about 10^12 trees in the world.
http://www.npr.org/sections/goatsandsoda/2015/09/02/43691905...
Good point. I'm usually so eager to mention that, too!
Trees regulate not only the microclimate, but the macroclimate as well. Isotopic analysis shows that about 80% of all terrestrial rainfall is caused by transpiration from trees and other plants. Trees essentially act as a giant inland irrigation network, using solar power to transport water via the airstream to power the water cycle. https://www.nature.com/articles/nature11983
So trees cause rain. This explains why deforestation causes desertification. I would also recommend the excellent Greening the Desert documentary for an example of the reverse process. https://www.youtube.com/watch?v=2xcZS7arcgk
As usual, it will get put off for lack of funding again and again until the old system finally breaks. Then there will be much action and much loss.
"The urban landscape accounts for 10.6% of England, 1.9% of Scotland, 3.6% of Northern Ireland and 4.1% of Wales. Put another way, that means almost 93% of the UK is not urban"
And of that 7% that is designated urban, a large proportion is gardens, parks, allotments, sports pitches etc.
Woodland (i.e. Forests) is 12.6% of the UK which is double the size of designated urban areas even if green spaces are excluded.
The majority of the UK is enclosed farmland. So to significantly increase the forestation of the UK, you would need to repurpose farmland.
Unfortunately, they are (at least in Scotland, which is the bit I know about) mostly just tree farms, growing very fast-growing, densely planted pines (Norway Spruce? can't find a reference), frequently in unsuitable areas, and the results were ecologically... poor. Nothing lives in them and they're alien trees anyway. The phrase 'trash forest' was coined.
These days there's a lot more awareness that this is bad, and they're growing more deciduous wood and relying less on monoculture, but it's still a problem.
Not the best picture, but here's a street view shot of one:
https://www.google.co.uk/maps/@57.5275519,-5.186435,3a,75y,2...
(Incidentally, if you look in the opposite direction; that island in the middle of the loch? That's native woodland. The slopes beyond are supposed to be covered with it. But it was all felled centuries ago, then the deer population rose, and now they're making it impossible for native forest to be properly established. You want to promote forest growth in Scotland? Cull the bloody deer down to about 10% of their current population. And then reintroduce wolves and lynx.)