We are really talking about three different markets with TPE-US and everyone plays to their strength and try to dominate their own niche.
DL and UA are competing on US originating fares with TPE as end points. If you are in a non-gateway location in the US, you can connect at SFO or SEA without much backtracking. Doesn't make a big difference on timing but FF program may make a difference. DL and UA don't need onward feed at TPE because they are absolutely not interested in chasing those fares.
US gateway O&D are dominated by the Taiwanese carriers except at SFO where UA also has significant O&D market share. In all other gateways, BR, CI and now JX are the preferred (and often only) choice for non-stop flights between US and TPE. And logically, no feed is required for Taiwanese carriers to dominate this market segment. It is by definition O&D only for travel between TPE and LAX/SFO/ONT/SEA/ORD/IAH/JFK.
And finally, US-Southeast Asia market is very fragmented. US airlines don't factor in very much between US and Southeast Asia - if you try to buy a ticket with US3, you end up on their JV partners via NRT or ICN instead of TPE. For Taiwanese carriers, since TPE is only one of many options to connect and they are locked out of US3 JV feed, it means they have to invest in marketing and distribution to generate corporate and VFR sales. BR has been far more active and successful than CI in marketing itself as a connection to Southeast Asia and has carved out a very significant distribution channel presence in both ends (US originating and Southeast Asia originating). This is basically the explanation for why BR has higher load factor at JFK than CI (BR has higher market share for NYC-Southeast Asia than CI).
As to why BR is more successful than CI in the US-Southeast Asia segment, I think it has to do with their respective historical market position in TPE O&D. CI used to dominate the O&D so BR had to get creative. The O&D market is more balanced now but BR still maintain its structural advantage in connecting traffic because of those early marketing and sales channel investments.