There is no theory of English syntax which contains all of the rules generating the class of English sentences, where the theory both explains why the rules are what they are, and uses just a few core principles. Such a thing is impossible in principle, since the thing that you're looking for isn't well-defined. The empirical domain is not well-defined for a number of reasons. First, "English" is too broad, so no well-defined system can simultaneously include and exclude sentences like "I might should go", which fyi is perfectly normal in some parts of the US. Other non-geographical examples are the variable acceptability of "I can has seconds?" as a question, and "If I was hungry, I would have eaten something".
It is also not clear from your question what you mean by "syntax", especially the extent to which you specifically exclude lexicon, morphology and phonology from consideration. As I understand it, minimalist syntax has relatively little by way of rules of syntax, and much of the work is shifted to the lexicon (also a feature of HPSG). Supposing we agree that *"I need" is ungrammatical, is that a fact represented in the syntax, or in the lexicon?
Somewhat finally, your concept of "explanation" needs to be clarified. The notion of "explanation" is notoriously elusive yet often-invoked in science. There is an equation in theoretical physics E=hf which states that the energy of a photon is the product of its frequency and Planck's constant. This does not explain why this particular thing should be the rule. I could assume that you are referring to Chomsky's third desideratum of theory (from Current issues in linguistic theory), namely "explanatory adequacy". This principle basically says that an explanatory metatheory allows a single grammar that generates a particular language. Explanatory adequacy can only be a requirement of a metatheory of language, not the syntax of a specific language. Since you are just asking for a grammar of English, then it is meaningless to require explanatory adequacy. And therefore, you must have in mind something completely different when you invoke the desideratum of explanation.
I think it is reasonable to conclude that we do indeed have a plurality of unrelated rules in English. For example, we have a passive rule (“The cat ate the mouse” → “The mouse was eaten by the cat”) and a raising rule (“The people believe that Tom is a thief” → “The people believe Tom to be a thief”). There are also rules for placing negation in a sentence. Unless you go with Greg Lee’s general purpose rule “a sentence is any number of words”, you are going to have a plurality of unrelated rules. What is in the syntax / lexicon dichotomy for us is that this kind of scheme might work, if you have a very rich lexicon that states in great detail in what context you can insert “mouse” – that is, you put pretty much all of the syntax into the lexicon.
I suggest taking the Aspects model seriously, as implemented in Burt’s red book From Deep to Surface Structure, since it is an actual set of rules in a concrete framework, even though it has a ton of problems. It at least gives you a concrete target to shoot at. There are separate rules of Passive, Dative, Agent Deletion, Tag Formation and so on. Objecting that these are all separate unrelated rules and thus mere descriptions lacking explanatory form holds linguistics to an unrealistically high standard of idealization – not even physics as managed to reduce all laws to a single statement. If you can show us an example of a “more explanatory” theory that actually has a single rule that subsumes Passive, Dative, Agent Deletion, Tag Formation into one rule, or declarative-formation and interrogative-formation in one rule, then we might be able to address your desideratum of explanation.
Your explanation of your concept “explanation” seems to depend crucially on the validity of “construction” as a fact of grammar: that suggests then that you’re actually looking for an implementation of English syntax in Construction Grammar, or perhaps Relational Grammar.