My recommendation? The TIVA C for $12 + shipping
I agree with your points about Arduino w.r.t. an embedded programming career.
I don't. Projects get larger and larger so you have to depend on libraries. Using a higher level programming language which avoids pointers, buffer, memory allocation, etc pitfalls makes a lot of sense to make code more robust. I think we'll see more and more higher level languages being used on microcontrollers in the future.
BTW going for the cheapest development board most certainly isn't a good idea especially when it serves as a dongle for the development tools.
Of course developers don't do everything from scratch, we use many libraries in our development as well.
But what happens when there is an error on a "tested" function because you are using it on a new generation of the product? are you just going to submit a ticket and pray someone fixes it eventually? I don't think the costumer would be to happy to know you have no clue were the problem is.
My point is... yes, normally you don't need to go deep down (unless you develop Drivers) but that doesn't mean you shouldn't know, because the fact is, you will need it eventually. Maybe you en up working for a company were you have to setup a project with a new chip an a custom BSP with external memory and other stuff like that, or maybe you end up doing a high level graphics Driver that requires for you to know about registers and how to access that peripheral. In the end, a good Embedded SW Dev needs to know the basics to debug the really complex problems when it is needed.
One of the best Embedded SW Devs that I've worked with knew very little about embedded and that was no problem... until he needed to develop a driver for a new peripheral
You could "survive" without knowing about low level, but in my opinion it doesn't hurt, because you will understand the system way better if you understand how it works!
And about going with the cheapest board. I really see no problem if you don't plan on doing you own HW, which based on the OP's question I believe is not what it is intended.
If you know cheap board with an ICD that has a free toolchain I would like to know (seriously) because I looked a while ago and I couldn't find one so I went for the Tiva C.