Hindsight is 20/20. At the time, the actual leading folks involved considered it a very, very serious threat.
Had England offered dispute arbitration (we were NEVER going to see Redcoats on the Rappahannock!), they likely would have couched it in terms of avoiding greater bloodshed, of interrupting a potential Servile Insurrection (with all its additional horrors), and as flexing their influence as 'the' final global arbitrating power. That's how their public excuses would likely run, anyway. A resultingly independent CSA would be no economic threat, would weaken the rising power of the competitive USA, would absorb both parties in their own internal revenges\expansionist goals, and leave Britain a free hand to dominate, manipulate, and profit from the result without the investment of a single soldier.
Any number of multiple outcomes are possible, of course. And that's my point. With quasi-perfect hindsight, we say our outcome was the ONLY one possible. We are blindly mistaken in our well-intentioned self-delusions.
To the folks at the time - the actual leaders facing the actual situations and making the actual calls - the threat was very real indeed.