I am curious whether it's just my brain that doesn't work right
You're describing your travel experience to someone... which of these sentences would you SAY? As in, speaking, not as in writing an email/etc -- you're at the office or somewhere talking to somebody. Assume that you actually have the experience of travelling to France and Spain at some point in your life, and someone's asking you about what countries you have the experience of travelling to or plan to travel to.
(oh, and assume appropriate abbreviation, ie, "I've" instead of "I have", if that helps.)
[Poll #1413618]
And a better question: does it change if you are writing a report or article rather than just speaking?
(oh, and assume appropriate abbreviation, ie, "I've" instead of "I have", if that helps.)
[Poll #1413618]
And a better question: does it change if you are writing a report or article rather than just speaking?

no subject
There are a decent number of things we learned (eg, adjくありません, 〜ませんでした,たくさん, はいお陰さまで) that almost never show up actual normal conversation or even writing, however, they start showing up much more frequently when in a keigo mode and are consequently not totally useless. Not sure if that counts.
IIRC we never learned 訪れる--it was always 行く. Likely because 行く (and Verbに行く) was far more generally applicable. However, I get the impression that standard Japanese English-teaching emphasizes memorizing lots and lots of words and additionally preferring specific, easy-to-translate words over general multi-use ones. (And goodness help them when they have to deal with prepositional verbs.)
Edit: Ack, I just realized that you compared it to 会う, not 行く. We did use 会う and never learned 訪ねる, but I'm not sure we ever used 会う for 'visit'--I think we were never in a situation where we needed to say we were visiting a person at their house and couldn't use 話しに行く or some similar construction. (BTW, おとずれる = to visit a place, たずねる = to visit a person. Same kanji, different okurigana.)
no subject
I had a student ask me last week why there are so many uses for the word "like". "Why isn't it just '好き'? I like steak?"
I guess the thing is, there are definitely some words that I feel like I learned incorrectly early on in Japanese, got the patterns in my head, and now they are fucking up my ability to say some things properly. Like we were taught ひどい means rude, I think... which is not true, the right word is 失礼. And apparently 会う is really "see" and not "meet".
OH! One big one is that we seriously never learned しゃべる for speaking but I swear that's actually what most people use for it instead of 話す or 言う. Like for "What did you say?" people seem to use 何をしゃべった? Things like that.
I dunno, though, it's hard to learn everything. The other day a girl said to me お家は? for "where do you live?" but my brain heard "an uchiwa?"
no subject
I don't remember being taught ひどい at all. We did get 失礼, but only in the set-phrase for entering/leaving a room. 会う is just a pretty broad word, and different of the kanji for it (会逢遇遭) have different connotations, and trying to map it to exactly one word in English that you can use for a guide is probably doomed.
Yeah, we learned 話す because it's the formal version of 喋る, under the "speak as inoffensively as possible" principle. I'd be extremely surprised if 言う doesn't show up at all ever, considering as how it's in so many linking and adverb phrases.