Closes https://github.com/ppy/osu/issues/30163.
If I'm to be blunt, the decoupled stuff in song select makes my head
spin. I spent a solid 20 minutes thinking how I was going to fix this
one but then finally realised that generally most of the cause there
was the fact that `AdvancedStats` was seeing the new rulesets *before*
the "ensure global selected mods are valid for current ruleset" logic,
and so decided to just _delay_ that until the decoupled transfer
thingamajig happens.
I was honestly considering combining `BeatmapInfo`, `Ruleset`, and
`Mods` into one property on `AdvancedStats`. I figured I'd rather not
push my luck and try the baseline version first, but I honestly think
that direction is going to be required at some point to properly corral
all of the decoupled madness taking place in song select.
After switching `UserLookupCache` to `GET /users/lookup` from `GET
/users`, multiplayer sort of breaks, since the former endpoint does not
return `ruleset_statistics`, which are used in multiplayer to show
users' ranks. Therefore, switch multiplayer to use the appropriate
request type directly.
Rather than control the propagation of the value between the slider and
the textbox, add a property that controls the propagation of the value
between the bindables inside the form control to external bindables.
This will help alleviate issues where the external bindable update
incurs overheads due to having heavy change callbacks attached.