Consider the following statement:
If I were you, I would go to the doctor.
As far as I know, a modal verb or a past tense can be interchangeably adopted to express the subjunctive mood. It seems, however, that most subjunctive sentences commonly prefer to use a modal verb, rather than a past tense. In conclusion, my questions are
Is the following sentence grammatically correct?
If I were you, I went to the doctor.
If so, what is the reason most people prefer to use a modal verb?
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
I wish you went to the doctor more often than you do.
...or..
If you went to the doctor tomorrow, I'd drive you.
...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).
It is better to understand conditional sentences with different moods associated with the condition and the proposition, as in French.
In French, the equivalent of “if I were you, I would do something” is “Si j’étais vous, je ferais quelque chose”. The condition is in l’imparfait and the proposition is in le conditionnel. It is important to not confuse the moods used in each part.
Comparing this to English, I would say the conjugation for the past subjunctive is always the plural past tense form or (for the past subjunctive perfect) the past perfect tense form, and the conjugation for the conditional is “would + infinitive” or (for the conditional perfect) “would have + past participle”. Sometimes one can replace “would” with other modal verbs, e.g., “could” and “might”. And the second conditional is “if past subjunctive, conditional”, and the third conditional is “if past subjunctive perfect, conditional perfect”.
It seems to me that it is always required to use a modal verb in the conditional mood.
There is also a present subjunctive mood in English. You may or may not use a modal verb in the present subjunctive mood, examples are:
