I respond!
A more accurate version of "i before e" that I've heard is "i before e, except after c, when pronounced [iː]." This works a little better (and excludes weird, which OED claims is pronounced [wɪəd], which is perhaps a little off, but it's definitely not [iː]) but creates more exceptions like friend, which obeys i before e, but is pronounced [ɛ] rather than [iː]. Ultimately though, anything you come up with is probably just going to be a fallible metric, as English spelling is idiosyncratic, with the spelling of words sometimes reflecting arbitrary facts about their etymologies and/or previous pronuciations rather than any kind of phonemic representation. For instance, in the case of weird:
Though a nominal form of weird has been around since Old English, e.g. "Hie wyrd forsweop on Grendles gryre" from Beowulf, where the meaning is something a bit like fate, the adjectival form is much more recent, and originally had a similar meaning to the old nominal form. This seems to have first emerged in Middle English, as in "A werde-sister, I wait neuir how" (1400ish), but it seems that it's Shakespeare that really got the usage going, and he (or perhaps the typesetters) spells it as weyard and weyward (some analogy with wayward was obviously going on). I guess the OE form was really just pronounced as one syllable with a dipthong, and with a monopthong for the form werde, but the meter of Shakespeare often suggests the word was bisyllabic - for example, try picking out the four iambs in "The weyward Sisters, hand in hand." The y was later changed to an i with an umlaut, indicating the bisyllabic nature, leading to weïrd, but the diacritic was dropped at some point, and the word often isn't bisyallabic these days. At some point vowel shifting led to the first vowel being pronounced higher in the mouth, more like or [ɪ] than the original [ɛ], giving us today's pronunciation alongside the spelling weird.
You can make sense of this in various ways, and it's a fairly sketchy account that probably has some inaccuracies (also it's incomplete - rhotacism, for instance, makes everything more complex), but it demonstrates the somewhat arbitrary nature of English spelling and how it's often unrelated to the phonemic form of words. This is a somewhat recent phenomenon - both OE and ME had more consistent and phonemic spellings, but influences like vowel shifts and import words have lead to the more scatterbrained nature of Modern English spelling.
It's the same story here - though both have OE forms (hÃehþo and wiht respectively - very different), various influences have led to different pronuciations but the same spelling, really as a result of chance. The height form was pronounced with an [eɪ] for a while (the same vowel Shorty transcribes as /ā/), which fits the spelling better, but analogy with high and Northen English uses led to a different vowel (today realised as [aɪ]), and for a while the two different vowel spellings - Milton for instance, uses highth; "To attaine The highth and depth of thy Eternal wayes," - but the ei spelling was settled on, probably mostly by chance. Weight seems to have gone the other way, starting out with more of an [ɪ] sound, but changing both in spelling and pronunciation through analogy with Old Norse forms with [eɪ] vowels and the verb weigh.
So yeah, it's all a tanged mess of etymologies, and not really the sort of thing most linguists tend to look at, as they focus more on generalisations than on specifics. Some people are more interested in this kind of thing though - I did all my research for this post using the etymology section of the OED. One final point: though English has a fairly crazy spelling scheme, it's by no means a total disaster - consistent patterns can be identified, although they change depending on the type of words you look at. Okay, it's not as neat as something like Spanish, but be thankful we haven't got a writing system like Japanese.
Note: I've used (very) broad phonetic transcriptions throughout so that I don't have to commit to phonemes, hence the square brackets rather than the slashes. If you don't know the sounds that the IPA symbols represent, Wikipedia has a reasonable guide.