So I would say engineer is a poor term, but also the only one we've got for now.
So it is not exactly like you've said:
> You can't design it first and then go and build it.
You can design, but you cannot build.
A process of building by a design can be paralleled with deploying software -- suddenly there is a hairy real world, not all the hair was considered at the design phase, and either we hack around existing software (i.e. design plans), or call a programmer to redesign.
You can't design it in full in one go, but you can design it and then incrementally update said design. Sadly many (companies) do not. But you can define the problem(s), the scope, the scale, and then design a solution appropriately to meet those needs (for a defined period of time). That's what distinguishes software engineering from hacking. They both have their place. Many companies claim to do the former but are mostly doing the latter. Software is still early in its life and as various kinds of system designs stabilize, so will the formalizations around what it means to be a software developer. Reading a book like Designing Data Intensive Application's you can't help but see those formalized topics budding.
As a designer I’m a bit embarrassed by the trend to describe ourselves as product designers. To me a product designer makes chairs and coffee tables.
The only reason you can design a bridge beforehand is because (millions?) bridges have been built before so you can apply the lessons learned. Even if your bridge is "unusual", it will still be similar enough to older bridges so you don't have to invent the vast majority from scratch.
Other kinds of engineering don't have the luxory of leaning on the prior experience so much, simply because there is less of it. SpaceX's reusable rocket could not have been be fully designed before built, simply because nobody built a reusable rocket before. But it could be done through iteration, which is just another name for experimentation.
Software tends to be less like bridges and more like rockets... all of which falls within the spectrum of "engineering".