I would argue that, grammatically, the "to" infinitives are not adverbial and they are not objects. They are verbs - infinitives that are serving different purposes, through different mechanisms.

1. I failed to persuade her.

From my linked article above, the purpose of the infinitive "to pursuade" in this case is to indicate the purpose or intention of an action. The action is that [I] the subject [failed]. What goal or purpose did "I fail"? The purpose was [to persuade] the direct object, which is [her].

[I] is the agent and the subject.

[failed] is the main verb, in concord with the agent. It can be both transitive and intransitive aka it is ambitransitive. We could say "I failed" and we could say "I failed her" - both of those are grammatical.

[Her] is the 3rd person objective pronoun. Grammatically, it is the direct object.

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EDIT

3. I want to go outside [from the comments]

I did some more research, and I found this webpage about the verb "want". I noticed something about "want" because of this site - "want" is an action. It's possible to both want an object (like in the sense of wanting to buy something), and to want to take an action (like wanting to walk). Grammatically, the information about who is doing an action will already be on the main verb. So if the action is to take another action, then the second action will be either in the "to" infinitive or the "zero infinitive". If there is a specific name for verbs that do this, though, it is outside my ability. This also might be relevant information for example (1), but I am reluctant to speculate much on it.

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2. The bomb is about to blow!

EDIT

I realized that addressing this example as a main verb and an auxiliary verb is more accurate grammatically. Thus, I edited my answer. Here is a quote about auxiliary verbs from Ginger Grammar:

Auxiliary (or Helping) verbs are used together with a main verb to show the verb’s tense or to form a negative or question. The most common auxiliary verbs are have, be, and do.

[The bomb] is the subject.

[is] is the auxiliary verb (it is a conjugated form of the verb "to be," but is being used in a special way in this case), in concord with the subject.

[about] is grammatically an adverb here, indicating nearly or almost.

[to blow] is the main verb.

The reasoning for this is that the main verb is the one which is the action of the subject. Since the bomb is the subject, the action to blow - simply existing ("is" as the main verb), is not the action in this case.

If we wanted to see this in present tense, it would be The bomb blows = the bomb explodes right now. Adding the present-tense auxiliary verb "is" grammatically indicates when the bomb blows up. In this case, The bomb is blowing = the bomb is exploding in this moment, and will continue to.

Lastly, adding the adverb "about" gives even more information about the state of the bomb. The bomb is about to blow = the bomb explodes very soon, but has not started to yet.

Short Answer

Neither object nor adverbial but complement.

Long Answer

There are many kinds of complements in a clause other than the object of a verb.

The to-infinitive construction itself such as to persuade her and to blow can be complement as well as adjunct (i.e., adverbial), depending on context.

In the given context of 1 and 2, however, it's not an adjunct. If the to-infinitive phrases were adjuncts in 1 and 2, you should be able to front them, but you can't:

a. ?To persuade her, I failed.

b. ?To blow, the bomb is about!

These are not objects either. In 1, the verb fail normally doesn't take an NP object denoting an action.

a'. ?I failed the persuasion.

In 2, the preposition about cannot take a to-infinitive as object. Prepositions only take the V-ing form as object.