Context free grammar (cfg) satisfies your requirements, in a certain sense, when interpreted in a certain way.
(1) Among transformational theories, it is Minimal, since it has no transformations.
(2) It distinguishes X-bar types from X types in a systematic way, by (at least in my interpretation) not having X types as primitives. To be consistent with traditional grammar, it is usual in cfg to write, e.g., P-bar -> P N-bar; P -> under, ..., but this is a bad move. It should be P-bar -> under N-bar, ... If we still want to refer to traditional parts of speech, we define: (a) a head is the pronunciation part of a context free phrase structure rule (cfpsr), and (b) a P is head of a P-bar. Similarly for V (head of V-bar) and N (head of N-bar).
(3) It is binary-branching, at least in the most straightforward way of doing derivations and displaying derivation trees. There is a rule of replacement, allowing a new cfpsr to be derived from two old ones:
replacement: From A -> xBz and B -> y, derive A -> xyz, where A, B are grammatical types (i.e. nonterminal symbols), and x, y, and z are strings of grammatical types and phonemes (i.e. terminal symbols).
A derivation tree is a branching diagram built up by joining subtrees each of whose mother nodes is a cfpsr derived by the replacement rule and whose two daughter nodes are the two cfpsrs used to do the replacement.
(4) It's pithy. Although I used -bar names above, since there are no non-bar primitive syntactic types to distinguish them from, it's not actually necessary to use the bars.
(5) There are no primitive transformations (other than the replacement rule), but derived transformations can be introduced as relations on the cfpsrs, following the method used in GPSG.