The reason why
- I love it when he visits and am looking forward to his next time
sounds starchy is that Conjunction Reduction is being called on to do too much.
Using a different metaphor, there are too many balls in the air to keep track of simply.
The full sentence, which is long but clear, is
- I love it when he visits and I am looking forward to his next visit.
The only reason why one might want to shorten this sentence is to save syllables or letters. Though there's no reason why one can't substitute the indefinite pronoun one for the last word -- it doesn't help anything, but it doesn't hurt anything, either, and it's speaker's choice in situations like that.
- I love it when he visits and I am looking forward to his next one.
So far so good. Another thing one can do is delete the repeated I via Conjunction Reduction, e.g,
- I like salsa and I dislike mayonnaise ==> I like salsa and dislike mayonnaise.
However, there are two problems with deleting I via Conjunction Reduction in the example sentence. One of the problems is that it leaves an uncontracted am in the second clause; this can sometimes cause problems.
The bigger problem, however, is that when searching for the subject of am, the parser needs to rewind the sentence back to the first syllable to find the I that's been deleted; and there already is another, different, subject pronoun he in the clause when he visits that's closer than the I subject the parser is searching for, but doesn't agree with am.
Depending on how one's parser is designed, this may result in more work than should be necessary for the parser, which may be irritated enough to load the debugger and stop processing.
The net savings in either syllables or letters is one; that's not much profit for all the processing work it causes. That's one of the features of starchy language -- it takes too much work for too little return.