Starting first with your question about why RD and not computers/tables.
First, you're taught tables to start (many hours spent comparing RD profiles to VPM and Buhlmann + GFs). You're also always encouraged to check planned profiles outside your RD comfort zone against software. Limitations of RD are of course discussed in detail in classes.
Second, as limeyx noted, RD is all about taking control of your diving. You're not merely reacting to the profile that some black box in a computer spits out in the water. Having to track your time and depth (for averaging purposes) keeps you engaged. The subsequent "math" to compute the shape of the profile is really trivial. Spending 100s of hours calculating average depth on recreational profiles before venturing deeper builds up the skill so it is second nature (you can compare the averages you generate in the water with later downloads while you learn this skill). Nor are you blindly following your buddy. Everyone tracks the dive.
Third, computers aren't providing the exact shaped profiles we want to dive. For example, they aren't providing the same deep stop pattern we choose, and don't emphasize high PPO2s at gas switches.
Fourth, as you note, the entire team would need the same computer (ignoring the third point for a moment). What happens when that model/algorithm is discontinued, or you travel somewhere new and people have a different computer? Nevermind the expense of a trimix computer compared to a BT. A similar issue of course arises for tables as well. At a minimum, the team (and the agency, if you want to make global unified diving a goal) must decide on a single set of tables for various depths/BTs/gases (both back and deco gases). RD is one such unified approach.
Fifth, RD allows for more flexibility than tables (unless you plan to cut a lot of tables...). Of course you enter the dive with a plan ahead of time, but RD allows for modification of that plan once in the water (within confined and understood parameters). Shorter/longer BTs, shallower/deeper depths are all easily accounted for (again, with understood parameters guiding these choices). This is particularly useful for exploration dives. Lost deco gases are also easily accommodated in this system, where new profiles can instantly be generated on the fly uniformly in the team. All of this can be done without having to pull out a set of tables.
As what to do if teammates disagree about the required deco, first, you have all of the deep stops and all the time at the gas switch to try to sort this out (here, wetnotes can help). If you're still unable to come to a consensus, just deco out on the more conservative schedule. Absolutely worst case, you can always deco on the max depth and total BT. In my dives, it's never been an issue.
Why do teams not use the same computer or table or agree upon the controlling one in advance? What happens when members come up with different computations using RD?