Short answer:
- It switches to were for "wish" statements because were is acting as a marker for such statements, rather than an actual past tense verb.
- Comparisons between middle/early modern English and today's English are tough to make, as no language is static. We've consistently been losing the case and inflection system from English for centuries (When was the last time you saw "Olde Shoppe" when it wasn't an obvious throwback?). As a result, what is proper during Shakespeare's time, and proper now, are two very different grammar codes for writing.
Long answer:
Most languages, especially older languages have a number of cases (nouns) and inflections (verbs) that English syntax has subsumed. For instance, in biblical Hebrew, A number of Greek variations, Latin, etc, the difference between the subject and predicate noun is clearly marked (not always so in biblical Hebrew, but still pretty common). Specifically to this question inflections changed in verbs creating the Subjunctive (possibility) and the Optative (slight possibility) tenses, in Greek; the Jussive (expression of command) and the Cohortative (exhorting a person to do something) in Hebrew; etc. In English, these have mainly fallen away, replaced with sentence placement and modals, though some are still dependent upon case-forms.
So, a sentence based on a modal and syntax might be, "We could go to the store today." It is based on syntax/position because this (admittedly stilted sentence) "Go to the store today, could we?" is now asking for permission, even though the same words are used without changes tenses.
For nouns, we can see it in: "Who is going to the store," and "You are going to the store with whom?" is an example of "who" changing case form—from subject to object.
For verbs dealing with subjunctives, were has become a condition marker for what is essentially the Optative case—remote possibility: "if he were to win the lottery, I still wouldn't go out with him." Don't think of it as past tense anymore. While it uses the past tense of to be in the plural, it's now a marker for a different function of language.
Why then, do we continue to use the present tense in some subjunctive sentences? That is dependent upon the action within the apodosis (second part of the conditional sentence). To use your sentences:
"It is essential that she be present."
"It was essential that she be present."
The past tense is expressed in the protasis. But the action itself was stative. You're stating a fact that has no ending. A past tense means that the action has been completed. Thus, "she was present" could be used if she walked into the room, thus completing the stative verb. However, it would no longer be a subjunctive, because a subjunctive is not completed action. The present tense must therefore be used to keep the action open.
At least, that's how I see it.
You're confusing traditional Latin grammar terminology with English grammar terminology,
and with modern linguistic terminology, as well.
Mood, Voice, and Tense were traditional inflectional categories of Latin verbs. I.e,
every verb in Latin was inflected (marked uniquely) for some mix of mood, voice, and tense.
Latin had six tenses (by a strange coincidence the same six you listed),
four moods (indicative, subjunctive, imperative, and interrogative),
and two voices (active and passive). That was Latin.
English has two tenses (Present and Past), no moods, and no voices.
In particular, English has no subjunctive mood, so you don't have to worry about it any more.
However, many other languages have rich inflectional systems, even richer than Latin.
Sanskrit and Greek both had a Middle Voice as well as Active and Passive, for instance,
and an Optative Mood (used for things one wishes and hopes for), and Sanskrit also had a Benedictive Mood (used for blessings).
And that's just Indo-European. There are lots of other ways to organize these matters.
Best Answer
It's important here to distinguish between the preterite form of a verb, and the past tense meaning that is normally associated with the preterite form. The modal preterite use of some English verbs indicates modality, as the OP points out (not past time). For example, in
...or..
...use of the preterite form of the verb (went) does not indicate past time.
To answer your question (why you can't say "If I were you, I went to the doctor"), English does not grammaticalize a generic concept of modality (nor does it have a robust grammatical category "subjunctive" as French does), but instead has several common constructions that can be used to convey different types of modality. The modal preterite construction cannot be used in the same situations as a modal auxiliary can. The modal preterite expresses modal remoteness, generally in one of the two types of constructions:
Auxiliary verbs can be used to convey a larger range of modal meanings (with different auxiliaries having different modal meanings associated), some of which include remoteness. In a sentence like "I would go to the doctor (If I were you)," would expresses two types of modality: propensity (lexically) and remoteness (preterite form).