There are a number of visual and audio settings that are commonly tweaked by competitive players.
Visuals
Many players in UT2004's old competitive community used the mutator UTComp to force a default player model + a brightskin. This is of course only available on servers using the mutator.
On servers not running the mutator, users can force a default model for all opponents via the User.ini file :
[XGame.xPawn]
PlacedCharacterName=Gorge
PlacedFemaleCharacterName=Rylisa
Only certain models can be forced, and you must play as one of models you force. I believe these are some of the available options:
Males: Gorge, Xan, Malcolm, Brock
Females: Rylisa, Tamika, Sapphire, Enigma, Ophelia
I know many competitive players who would set lower their texture quality to the very lowest, to reduce clutter. I don't believe this was allowed in tournament play, but that would vary by tourny rules.
Audio
On servers with UTComp, many players use CPMA style (damage dependent pitch) hitsounds.
Almost all players turn off their music, and some adjust their ambient volume and roll-off settings in UT2004.ini :
[ALAudio.ALAudioSubsystem]
AmbientVolume=0.125000
RollOff=0.400000
As you increase ambient volume, you can hear locational sounds such lifts and creaky boards more clearly, but there is also more environmental noise such as thunder, water, etc. So, it's a trade-off.
Reducing roll-off is likened to cheating, because decreasing the value decreases the dissipation of sound over distance; sounds travel farther. Competitive servers often use a 'lock roll-off' mutator that limits the value to 0.4 (default is 0.5.)
Other
Check out TweakGuides' UT2004 guide for additional information on configuring your .ini files.
Best Answer
This might be speculative in nature, but consider any other single-player sport that has teams. Take chess for instance. Despite being a 1vs1 game (usually), there exists several teams worldwide. Usually they start in schools, although one usually joins a federated team to access the local/global ranking system. Also, the following reasons apply:
In professional settings though, where representation to a wider audience may be required, scale mechanics may come in play, as it is cheaper for a single agent or representative to talk for a group than to have one agent per player.
Although I don't have experience with Hearthstone (or professional e-sports, by that matter), you can see how having more than one player on a easily reachable structure is beneficial to the player and to the group.