Я не хотела возражать, но мне всё-таки пришлось доказать, что задержка была не по моей вине.

Because пришлось is used in an impersonal construction.

In Russian, прийтись / приходиться + infinitive often means to have to, and the person affected goes in the dative case:

• мне пришлось = I had to
• тебе пришлось = you had to
• ему пришлось = he had to

So мне does not literally mean to me here in an English sense; it marks the person who was forced by circumstances to do something.

That is why Russian says:

• мне пришлось доказать

not

• я пришлось доказать

There is no normal subject in this construction, so the verb appears in the default impersonal form: пришлось.

Here пришлось means something like:

• had to
• was forced to
• ended up having to

It comes from прийтись, and in this kind of sentence it usually suggests that the action was necessary because of circumstances, not because the speaker wanted to do it.

So мне всё-таки пришлось доказать implies:

• I did not want to do it,
• but in the end circumstances made it necessary.

This is slightly different from just я должна была доказать, which can sound more like an obligation or duty. Пришлось often feels more like I had no choice.

This is mainly about aspect and nuance.

• хотела is imperfective past: it describes a state or attitude — I didn’t want to
• возражать is also imperfective: it refers to the general action of objecting, arguing back, or contradicting

So не хотела возражать means:

• I didn’t want to object / I wasn’t inclined to object

If you said не захотела возразить, that would sound more like:

• she refused to make a specific objection
• she did not decide to object once

That is more tied to a single completed act. The original sentence is more about general reluctance, so не хотела возражать is the natural choice.

Всё-таки adds the idea of:

• all the same
• nevertheless
• in the end
• after all

It shows a contrast between what the speaker wanted and what actually happened:

• Я не хотела возражать = I didn’t want to object
• но мне всё-таки пришлось доказать... = but I still / nevertheless had to prove...

So всё-таки emphasizes that despite the speaker’s reluctance, the action happened anyway.

Without всё-таки, the sentence would still be correct, but it would lose some of that sense of reluctant inevitability.

Because the sentence focuses on the result of the action.

• доказать = perfective, to prove successfully
• доказывать = imperfective, to be proving / to prove in general / to keep trying to prove

In this sentence, the important point is that she had to establish the fact — to produce the proof successfully. That is why Russian uses доказать.

Compare:

• мне пришлось доказать = I had to prove it, and the idea is the proof was achieved
• мне пришлось доказывать = I had to be proving / I had to argue and present proof, with more focus on the process

So the original version highlights the completed outcome, not just the effort.

Because задержка is a feminine singular noun, and in the past tense Russian verbs agree with gender and number.

• masculine: был
• feminine: была
• neuter: было
• plural: были

So:

• задержка была = the delay was

Even though English uses was for everything in the singular, Russian past tense shows gender.

This is the expression по вине + genitive? Actually in modern grammar it is usually treated as a fixed expression with по and a noun in a form that historically aligns with the dative. In practice, learners should memorize по моей вине as a set phrase meaning through my fault or because of my fault.

Here:

• вина = fault, guilt, blame
• моей = my, feminine singular, agreeing with вине
• вине = the required form in this expression

So:

• по моей вине = through my fault / because it was my fault

This is a very common Russian way to assign or deny responsibility.

Because не по моей вине is a very natural idiomatic way to say that something happened not through my fault.

There is a difference in emphasis:

• это не моя вина = it is not my fault
  This is a direct statement about blame.
• задержка была не по моей вине = the delay was not due to my fault / not through my fault
  This connects the event specifically to its cause.

So не по моей вине sounds especially natural when talking about an event and denying responsibility for that event.

Yes, but the nuance changes.

• не по моей вине = not through my fault, not due to blame on my side
• не из-за меня = not because of me

Не из-за меня is more direct and conversational. It focuses on whether the event happened because of the speaker.

Не по моей вине sounds more idiomatic when discussing responsibility, and often a bit more formal or careful.

So in a sentence about proving that a delay was not the speaker’s responsibility, не по моей вине is the better fit.

Возражать means to object, to argue back, to contradict, or to express disagreement.

It is often used when someone says something and you respond against it.

For example:

• Я не возражаю = I don’t object
• Он начал возражать = He began to object

In this sentence, не хотела возражать suggests the speaker did not want to argue, push back, or contest what was being said — but later had to defend herself by proving the delay was not her fault.

So it is not necessarily a loud argument; it can simply mean to raise an objection.

The word order is natural because it presents the sentence in a clear progression:

1. Я не хотела возражать — first, the speaker’s attitude
2. но мне всё-таки пришлось доказать — then the unwanted necessity
3. что задержка была не по моей вине — finally, what had to be proved

Russian word order is flexible, but not random. This order sounds smooth and logical.

You could move some parts for emphasis, but the tone would change. For example:

• Но мне всё-таки пришлось доказать, что задержка была не по моей вине.
  This puts more immediate emphasis on the necessity.

• Я не хотела возражать, но мне пришлось всё-таки доказать...
  Also possible, but всё-таки is often placed before the verb phrase it modifies, as in the original.

So the original order is a very natural, neutral way to say it.