I've been in the bike industry here for almost 10 years and offhand I think of about 10-15 other foreigners who are based here in Taichung working full time for bike companies in various roles. There are doubtless a few more than that island wide and lot more who aren't based here but visit as required to liaise with suppliers or co-workers. Honestly said, of all the foreign folks I know who work in the industry, mostly industrial designers, sales & marketing managers, engineers, sourcing & purchasing managers, business owners/entrepreneurs, product/brand managers and even riders/journalists, I can't think of one who was hired primarily because they could speak Chinese and in fact I know quite a few who can't speak Chinese at all.
I think the bike industry is no different to any other in that regard and holds true to the general law that you can do anything you like in Taiwan, but unless you've got related skills and experience that command a higher pay bracket in that field, you'd probably do better teaching English. Reason being that if even "very good" Chinese ability is the main thing you're bringing to the table, you're up against any number of local folks who have first language Chinese, "very good" English ability and are willing to start at NT$30k or less a month. That's not to say Chinese ability is worthless, depending on what kind of bike company you're talking about working for about and in what capacity Chinese would certainly be an asset, and in some cases a necessity, but IMO it's not going to land you a job outright, except in the kind of 'foreign face and a pulse' roles which are usually neither long term nor well paid.
The catch 22 is like most niche industries, only related skills and experience count, so without any it's hard to get your foot in the door, rinse and repeat. The bike industry, even globally, is a fairly small pond with a limited number of fish and naturally who you know can be more important than what you know (the same as anything else I guess). The big international brand players that you mentioned tend to fill "foreigner positions" in Taiwan either through word of mouth (bike industry folks do seem to jump from company to company a fair bit and talented folks are often headhunted away) or through their HR departments in their home countries as Taiwan is no longer seen as the kind of "hardship posting" it once was so more folks are willing to work abroad these days and transfer to TW from within the company. In these big multinational companies English is standard and there is often little or no need to speak Chinese since their TW employees and suppliers are used to handling everything in English that needs to be. Basically to get on board with an A-level international bike brand/company you need related industry experience and qualifications, period. Where there is perhaps some opportunity is the kind of smaller Taiwan "off brand" components makers and OEM/ODM/assembly factories who could actually benefit from a foreign sales/marketing/editing person who's Mandarin savvy and can put a little polish on their international image. That obviously takes a skill set beyond just Chinese and even then, the tricky part is that the successful companies in this category generally either already have someone in that role or are doing fine on their own, while the less successful mid-pack companies are often either not willing to offer anything beyond local entry level salaries and/or are totally in denial that they need or could benefit from the help in the first place.
Bottom line, at least in my opinion, Chinese ability won't open many doors on it's own, but it certainly won't hurt and neither will being a rider yourself and being passionate about cycling. In fact being a keen rider and being involved in the local cycling scene in TW may *perhaps* put you in a lucky break position where the 'right time right place' effect with the added bonus of your Chinese skills may lead to an opportunity, so at the very least being those two rungs up the ladder is a worthwhile start.
Sorry if this post comes across as excessively pessimistic, it certainly isn't my intention to rain on your parade or scare anyone away from trying to get into the bike industry, this is just one guy's opinion after all and other's MMV. If cycling is something you love to do and a career in the bike industry is what you really want, there are certainly worse ways to start out than learning Mandarin in Taiwan and trying your luck.