pût is subjonctif imparfait.

The subjonctif imparfait is used to tell the story, actions that advance the story.

You can’t use subjonctif présent.

The present subjunctive is used to express present actions or ideas which are subjective. It is nearly always found in dependent clauses introduced by que or qui, and the subjects of the dependent and main clauses are usually different.

I suppose I should have done this quick research before posting a question.

Apparently, the replacement is permissible because, according to the sources below, the replacement of imperfect subjective with present subjective is always permissible.

I might add that French subjunctives, then, seem to behave differently from English or German subjunctives but rather like English indicatives, in the following sense.

English and German subjunctives are unaffected by the tense of the main verb in whose context they occur. For example:

It is required that the child be accompanied by an adult at all times.

It was required that the child be accompanied by an adult at all times.

At least in formal writing, French subjunctives change to "match" the tense of the main verb as do English indicatives. For example:

Je préfère que vous arriviez à temps.

Je préfèrais que vous arrivassiez à temps.

I am posting this answer so any error in my understanding can be pointed out.

Sources

Wikibooks:

The subjunctive imperfect is very rarely employed in French; generally it only appears in literature and is viewed as archaic. It can in all instances be replaced by the subjunctive present. The subjunctive imperfect is employed in any instance in which the subjunctive is required, provided the trigger verb is in a past tense. In the example "Il fallait que le garçon allât à l’école", the subjunctive trigger verb "falloir" is in the imperfect, thus "aller" has been conjugated in the subjunctive imperfect. French speakers would normally express this as "Il fallait que le garçon aille à l’école", where "aller" has been conjugated in the present subjunctive.

French.about.com:

The imperfect subjunctive is used in a subordinate clause when the main clause is in the past. Its non-literary equivalent is the present subjunctive.

Indeed the subjunctive past tenses are nowadays only used in litterature and have become kinda obsolete in oral. Whereas the other romance languages have kept using them (Spanish, Portuguese, Italian people use them naturally in everyday’s life) the subjonctif imparfait and subjonctif plus-que-parfait are never used when speaking, so that they are even ignored by some French people, but knowing those tenses is an obvious proof of your knowing of the language! So yes the only tense used orally with the subjunctive mode is the present, even in the cases where imperfect subjunctive would fit better.