*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.<sup>&dagger;</sup> 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.

***
<sup>&dagger;</sup><sub>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. </sub>