Author here. Just for context tinyc.c was created in 2000 (I found the file in my archives and the last modification date is January 12, 2001). I was not aware at the time of Fabrice Bellard's work which after all won the IOCCC in 2001, so the confusion with TCC was not intentional. My tinyc.c was meant to teach the basics of compilers in a relatively accessible way, from parsing to AST to code generation to bytecode interpreter. And yes it is the subset of C that is tiny, not a tiny compiler for the full C language.
Haven't looked at what it would take to support C89 (or other) fully. Certainly it is possible to extend the compiler in small increments to implement more stuff: all the operators, local variables, arrays, pointers, functions, etc. The hardest part I would say is implementing types (starting with the C syntax for types!) and also handling the memory allocation for them. Nothing too complex when you know what you are doing.
My personal interest is with higher-level languages and I have worked on various Scheme systems over the last 30 years. If you like small language implementations then take a look at Ribbit which is feature-full with a tiny VM. Also check out sectorlisp (not my work) which is a fascinating tour de force.
tinyc.c was designed to illustrate three things at once in a second year course on concepts of programming languages. The course had 4 parts: imperative programming (using C), functional programming (using Scheme), and logic programming (using Prolog), and programming language implementation. For the imperative programming part it was important to show enough of the C language for the following operating systems course, so we needed to show C manual memory management and pointers in a rather detailed way. So tinyc.c was principally an example of programming with pointers, including pointer arithmetic such as *pc++. It was indirectly an example of compiler and interpreter, subjects we also covered in the course. I have also used parts of the compiler in a third year compiler course but not as the basis of a project. I have always asked (forced?) my students to use Scheme to implement their compiler projects... a much friendlier language than C for compiler writing.
I'm not sure what you mean by "minimal set of instructions". What is more important for bootstrapping is to restrict the programming style used to write the compiler. So if that was the goal I would implement pointers and maybe arrays, and not implement other types such as structs (which can be useful but add complexity to the compiler, i.e. it is no longer the case that a value fits in a machine word). Functions and function call would obviously be useful to add. But that's about it.
What I'd love to do someday is to write a compiler for a fairly complete C subset in POSIX shell. The goal would be to use only a POSIX shell and this compiler to compile TCC, and then use TCC to bootstrap gcc, all from source. This would be a great tool for reproducible builds from source. If someone here finds this interesting and would like to help out, please reach out to me.
