Why is the idea of Britain intervening in the ACW so common here? On the side of the CSA even more so(1)? Britain had nothing to gain by intervening on either side(2) - even more so by intervening on the side of the CSA which would be supporting slavery(3) (illegal for decades in Britain), whilst also breaking up a major trading partner who they were actually on decent terms with(4), with no territorial gain(5).
A few reasons:
1) Foreign intervention is perhaps the most plausible and reasonable chance of a Confederate victory that doesn't cripple it completely in the post-war world.
2) In terms of power politics not exactly, the British at this time were rather rightly worried about complete American dominance of the continent and potential expansion into the Caribbean and the Pacific which would threaten British interests in both those regions.
3) Technically yes, but in terms of the balance of power and a broader geopolitical situation, no. Until 1863 the war is not seen as anything but an internal issue about preserving the Union and until the passage of the 13th Amendment it isn't even about
ending slavery as a whole.
4) Cordial is probably the better word. Neither side had a great interest in another war and little to gain from it. There were tensions and war scares to be sure, but the lack of will to follow any of it prevented war. However, Britain can damage a potential rival while not seriously disrupting her own trade and economic situation. The US depending far more on British imports than the other way around and any brief interruption caused by a war wouldn't be damaging to the British.
5) That depends on the peace treaty. The San Juan Islands fall into British hands here, there is no chance of an Alaska Purchase for the US post war, Hawaii is not going to be annexed by the Americans, and some minor border issues on the Pacific end will probably be cleaned up in the British favor.