Hmmm....I think this depends largely on the quality of the CI the student gets, and what the student's goals are.
With someone new to CI-based instruction, the "C" is usually missing or incomplete. The student gets lots of input -- but input alone isn't sufficient (or else immersion would work more often and more quickly -- for many, it doesn't work at all.) The new CI-based instructor is also usually hesitant about giving grammar popups, because they're not CI. He won't repeat enough. He'll speed up without realizing it. It takes a heck of a long time to train yourself to stay slow enough for a beginner or a false beginner with really poor mastery of vocabulary (which describes many Taiwanese learners at the adult level). (Mastery, not knowledge.) The new CI teacher (if he's using TPRS) is hesitant about going into class without a script, so opportunities for things to just go the way they will go are missed. He will not notice chances to spend an hour expanding on a single phrase that came up and which could have led to all sorts of details and offshoots in the story (which means more CI, broader structure presented, and more repetitions of the key items). You don't get the slow development of the class culture over time, peopled with all sorts of characters that are the product of the collective mind of you and the students and won't be duplicated with any other group (and don't try, it doesn't work.)
And a big piece of the puzzle is the student's goals in studying English with a tutor. If it's test-driven, most likely a traditional teacher will serve the student's needs better (especially if he has experience with the test in question) since tests are by and large written by traditional teachers and aimed at students taught traditionally. Proficiency tests are the exception, IF they are really proficiency-based and not just labeled "proficiency".
I would prefer for my language teacher -- assuming I'm doing a language to become fluent, not for a test:
1. Someone experienced with CI (=reliable 100% comprehensible input). Luxury. Sit back and enjoy the ride.
2. Someone trying hard to improve his/her CI teaching (=attempt at 100% comprehensible input; I can help by signaling when I don't understand. Some frustration potential but I can coach or cue as needed.)
3. A really lazy teacher doing traditional teaching, preferably without preparation (=I can get him off track and force CI from him without his knowledge). He won't be solid on his grammar knowledge, so he won't overwhelm me with it, but he'll probably be able to tell me what the verb ending is for one form or another when I think I've noticed something, or how to say this or that.
4. A "good" traditional teacher (this will be an unbearable experience from my perspective, and indeed I had to walk out of the last such class I had. I just sat there thinking, "I just want to hear some simple language and understand it. is that so much to ask? You're a native speaker!")
I'm coaching a reformed grammar Nazi teacher right now who really wants to make CI work teaching a native American language. Every once in awhile, we're just off to the races with a long grammar digression. There's just no stopping it. But at the end of the session, he'll think about what he did and say, "Oh, that was too much talking about grammar." The disconnect in the class experience (I'm learning the language along the way) is huge for the students-- the other day I had the lovely feeling of "I'm understanding this! I can get this! I feel like I can even answer in [language]!" and then bam...grammar digression.
If a student asks for a grammar explanation, you provide it -- in five seconds. You should be doing lots of these on your own anyway, any time some interesting feature comes up that you want to highlight in that stage of the student's learning. I think that to "sell" TPRS to a group of Taiwanese adults, you would need to explain (briefly and persuasively, of course) what the idea was, probably in Chinese, first, and then get them to try it. There are some good demonstrations that can be used to let people really feel what it is to be dysfluent in a language even though you "know" it. Maybe we should get together and think hard about CI marketing, and produce something like a short DVD or a pamphlet or something for this sort of thing.
The main thing with TPRS is that people have to admit they have a problem. (This is starting to sound like a 12-step program!) Hi, I'm Mrs. Chang, and I'm not fluent after 15 years of English classes. Then they have to surrender to the higher power of the Brain (not the teacher, the Brain) which will do its job if it's given what it craves -- meaning it can link to sounds over and over.