(1) We had a great time visiting with my brother and family yesterday.
[. . .] my question is whether the persence/absence of the preposition "with" in (1) (or in any sentences) is optional without changing the meaning and the grammaticality/acceptability of the sentence?
Sorry for the delay. It is perfectly normal for "visit" to be used intransitively, as well as transitively, in American English. I wasn't aware that the British didn't use it intransitively. There is nothing wrong with that usage.
There is, however, a difference in meaning. When we use the intransitive "visit with," as in (1), or simply "visit" with no direct object or prepositional phrase following ("We visited awhile"), the meaning is essentially "talk with."
When "visit" is used transitively, on the other hand, as in "We visited them yesterday," the meaning is that we went to see them. For example, you can visit people in the hospital or in prisons. Prisons have designated days for visitors.
Here is what Bryan Garner has to say about this topic:
visit, vb.; visit with, to. To visit someone is to call on that person, especially as an act of friendship or courtesy. To visit a place is to go there to see (as a sightseer might) or to oversee (as an official might). To visit with someone, in AmE, is merely to talk with that person for more than a moment, usually in a personal way (at a watercooler, in the front yard, on a city street, in a living room, etc.)
- Garner, Bryan A. Garner's Modern English Usage (4th Ed.), p. 948. Oxford University Press: New York, 2016.