> By that logic I can slice open a sphere and call it a sheet

You can do this. If you remove a point (or a line, or really any connected component), you get a space which is the same as the plane. What happens if you remove two distinct points? You end up with with a very thick circle. Three points? It starts to get harder to visualize, but you end up with two circles joined at a point. As you remove more points you will get more circles joined together. From a mathematical perspective, these spaces are very different. If we start to allow gluing arbitrary points in the sphere together it gets even worse, and you can get some pretty wild spaces.

The point of surgery is that by requiring this gluing in of these spheres along the boundary of the space we cut out, the resulting spaces are not as wild - or at least are easier to handle than if we do any operation. To give an example, one might have some space and we want to determine if it has property A. The problem is our space has some property B which makes it difficult to determine property A directly. But by performing surgery in a specific way, we can produce a new space which has property A if and only if the original space did, and importantly, no longer has property B.

For property As that mathematicians care about, surgery often does a good job of preserving the property. In contrast things like just cutting and gluing points together without care will typically change property A, so it does not help as much.

> Likewise I wonder why we need to import a sphere rather than just pinch the ends of the tube shut and say it's now a sphere.

I am not an expect on surgery, but I think from a mathematical perspective, pinching the ends of the tube shut and gluing in a new sphere would be equivalent operations. This pinching operation would be formalized as a "quotient space", and you can formalize the sphere as a "quotient" space equivalent to the pinching.

> I don't really get the whole surgery concept, if you can just cut the torus (and this could be anywhere, eg along the length of its inner or outer diameters) and glue the edges together aren't you just waving away the problem? By that logic I can slice open a sphere and call it a sheet, or conversely declare all shees to be spheres that haven't been glued together yet.

It's admittedly been a good 15 years since I cracked open a topology textbook, but the high-level, hand-wavey idea behind this sort of topological surgery isn't that you slice up manifolds and glue them together willy-nilly, but that you do so in very precise and controlled ways, which (and this is the very important bit) you've proven ahead of time preserve (or modify in a knowable fashion) some property of the manifold you care about. Rather than waving away the problem, you're decomposing it into a set of simpler ones which are ideally more tractable for some reason or another (perhaps you can compute a given property from first principles for a sphere, but not for a torus, for instance).

The wikipedia page on this stuff (https://en.wikipedia.org/wiki/Surgery_theory) is quite technical, but you might be able to squint at it and get a sense for how it works.