This is not an AmE or BrE issue at all.
In English, we say:
I like it here. or I like it there. to mean: I like being or living in some place.
If you are in the place (city, town, area, region, country, etc.), you say: I like it here. If you are not, you say: I like it there.
This is basically idiomatic. The it stands for: living or being in a place.
Also, yes, in English, we say: I visited a place [city, town, region, country) and also say,for example, for that (tourism): I went to [some place] last year.
There is nothing wrong with saying "visit a place" (visit here or there) in English and again, there is no difference between AmE and BrE.
[the verb like has to be followed by an object (verb or object) unless it is being used with the function word to.]
I like being/living here. = I like it here.
I like being/living there.= I like it there.
It replaces the implied being or living.
I like playing piano. I like it. Playing piano. If I say: "I like it here" that means: I like playing piano here. [in this place].
Visit cannot be followed by anything but a place or a noun that stands for it: I visit here [noun] every year. I visit there [noun] every year. I visit it [the city, noun] every year.
like can be followed by a verb = I like living or being here. being or living here or there can be replaced by just it but you must keep here or there, otherwise the meaning changes.