stdint.h is nearing 20 years old now, I don't think it counts as a "newfangled thing" anymore.
But yes, there is plenty of very obsolete software still out there.
In the world of C, it is. Custom compilers for particular embedded systems, such as MCUs, often only speak C89 and nothing else. C99 still hasn't found complete and widespread acceptance.
Because sometimes you are not the author of the software you want to port. Of course, when you start porting the software, you'd use a (predefined) typedef.
You need some low-level justification for using an uint32_t and such: like conforming to some externally imposed data format or memory-mapped register bank.
The justification for <stdint.h> is that it's better to have one way of defining these types in the language, than every program and every library in every program rolling its own configuration system for detecting types and the typedefs which name them. Let's see, for calling Glib, we use guint32, for OpenMax we use OMX_U32, ...
Funny how these situations persist almost 20 years after stdint.h was standardized (and a number of more years after being draft features).
An expression of type int doesn't undergo any default promotion to anything, period; the widening promotion is only applied to the char and short types.
An int operand may convert to the type of an opposite operand in an expression, and an int argument or return value will convert to the parameter or return value type. That applies to int32_t also.
Anyway, you really have to know the rules to be working in C. Someone who uses fixed width types for everything doesn't know what they are doing and are just taking swipes at imaginary ghosts in the dark out of fear.
I'm guessing you've never ported code that was written for 32-bit ints back to a 16-bit architecture.
Always better to make your type sizes crystal-clear to the reader, IMO, even if it risks using a suboptimal word size on some other platform down the line.
This is backwards. You should use uint32_t etc. like any higher-level language would, unless you have specific reasons to know you need a machine-sized type. Making your code behave differently in some arbitrary, untested way on machines with different default int sizes isn't going to make it "less limited", it's just going to make it broken.