The tunnel was started in the 80s. Margaret Thatcher wasn't keen on massive public spending. See also another big Kent infrastructure project, the QE2 Bridge - en.wikipedia.org/wiki/Dartford_Crossing
I would say @irresistibleoverwhelm covered this quite well. The Chinese can do government funded projects like this, but in general, no, most other places would use private or a mixture of public/private. For example, Hinkley Point, although that's perhaps how not to do it - www.theguardian.com/news/2017/dec/21/hinkley-point-c-dreadful-deal-behind-worlds-most-expensive-power-plant
Getlink/Eurotunnel did go bankrupt a few years ago. I think if it can't be refinanced there's a provision in the treaty of Canterbury for the company to be replaced by another entity chosen by the British and French governments. I am pretty sure it's all covered by that. It's too strategically important for it not to be governed by international law.
Capitalism means that private companies own the means of production. As the revenue generation here is passenger/freight traffic, Get Link is not going to stop people using the tunnel.
If you're interested in how things work, take a look at another treaty, the treaty of le touquet, which allows the British and French to carry out border checks at the point of departure. In other words, you go through the French border in St Pancras/Ebbsfleet/Folkestone and the British one in Coquelles/Lille/Gare du nord. This is brilliant from a customer and practicality perspective, you drive off the train and straight onto the motorway, or leave the train at the gare du nord and are out in minutes, but Brexit made things a bit tricky.
The geography of Dover/Folkestone doesn't make it possible for large holding areas to be built there. That's why there's a massive lorry park up the road in Ashford.
The whole county and road system in Kent is funnelled towards the channel and it is brilliant to have a crossing that isn't weather dependent.
Strikes or storms mean Stack or Brock on the M20 which is miserable. That's another reason a bridge isn't practical. You need a constant movement of traffic otherwise things get difficult very quickly.