HAVE been is NOT "more commonly" used in the sense you describe. It is ordinarily a simple copula.

You may be confused by ellipsis here, and the dual senses to may have.^† In the perfect idiom HAVE been to X, meaning "HAVE gone to X and returned", the to X is a locative PP headed by the preposition to:

I've been to London to visit the Queen.
I've been to Princeton for a conference.

In the example you give, "I've been to visit my mother", the locative is omitted because it's recoverable from context, semantically overlapping the marked infinitival of purpose to visit. We infer that the to destination is the mother's residence. Ordinarily, however, the locative is required to "constitute" the idiom.

In any case, I don't think any native speaker would even momentarily take has been to mean "The effect has gone [somewhere] and returned". The idiom ordinarily licenses only 'agentive' subjects: an effect is not capable of "going" somewhere. Similarly, has awakened cannot be substituted, because an effect is not capable of "awakening" the base of a party: the awakening is the effect.

Consequently the default parsing of The effect has been is as the simple copula, and what follows is readily taken to be an infinitival predicative complement.

^†_{I suspect from your title question that you may also be distracted by the German use of be as a perfect auxiliary with verbs of motion. English used to have the same construction, but this fell out of use 300 years ago.}

HAVE been is NOT "more commonly" used in the sense you describe. It is ordinarily a simple copula.

You may be confused by ellipsis here, and the dual senses to may have.^† In the perfect idiom HAVE been to X, meaning "HAVE gone to X and returned", the to X is a locative PP headed by the preposition to:

I've been to London to visit the Queen.
I've been to Princeton for a conference.

In the example you give, "I've been to visit my mother", the locative is omitted because it's recoverable from context, semantically overlapping the marked infinitival of purpose to visit. We infer that the to destination is the mother's residence. Ordinarily, however, the locative is required to "constitute" the idiom.

In any case, I don't think any native speaker would even momentarily take has been to mean "The effect has gone [somewhere] and returned". The idiom ordinarily licenses only 'agentive' subjects: an effect is not capable of "going" somewhere. Similarly, has awakened cannot be substituted, because an effect is not capable of "awakening" the base of a party: the awakening is the effect.

Consequently the default parsing of The effect has been is as the simple copula, and what follows is readily taken to be an infinitival predicative complement.

^†_{I suspect from your title question that you may also be distracted by the German use of SEIN as a perfect auxiliary with verbs of motion. English used to have the same construction, but this fell out of use 300 years ago; in Present-Day English only HAVE is permitted as a perfect auxiliary.}