For what it's worth, if I pronounce "Tamsui" (a spelling that I hate!) as dumbly and fresh off the boat as possible (t-am-se-uh-ee), it's surprisingly close to the pronunciation in Taiwanese.
Hanyu pinyin does have to be learned, but, well, so what. That's how languages work. If you're going to spend any time in France or Germany or Italy you're expected to learn phonetic symbols have different meanings in different countries. If you're going around France saying "deh-uchs bee-air-ess sill voh-uhs plah-itt", that's on you, not the language.
Is it possible to have a written language that would encompass all the local languages? I'm not sure how that could work - it'd be like writing "Germany", with English speakers knowing that pronunciation, and Chinese speakers knowing that "ger-" = de, and "many" = guo; and Germany speakers I guess knowing "Ger" = deutsch, and "many" = land. If there's always a one-to-one consistent correspondence between the syllables in Mandarin and in Taiwanese and in Hakka and in all the different aboriginal languages, OK, I guess it's theoretically possible, but I'd be surprised.
Or do you mean multilingual signs, with Chinese characters in one line, followed by Mandarin pronunciation (Romanization system yet to be determined), followed by Taiwanese pronunciation (ditto), followed by Hakka pronunciation (ditto), followed by X aboriginal languages (ditto)? That'd be kind of cool - it's not unusual in Canada to see highway signs with English / French / local First Nations language.