Actually, rigid body simulators using impulse based constraint solvers can simulate soft constraints (including soft contact), and you can control the stiffness from very stiff to very soft without becoming totally unstable. See this slide deck by Erin Catto: https://box2d.org/files/ErinCatto_SoftConstraints_GDC2011.pd...
The only issue with iterative constraint solvers is the 'long chain' problem, where it's not possible to solve for longer stiff chains of constraints. Direct solvers can do it, but not with reasonable performance/memory usage.
Indeed, you can add a Featherstone articulated body algorithm implementation for such cases, or direct solver with good performance (see Maciej Mizerkski talk on Roblox solver here: https://youtube.com/watch?v=P-WP1yMOkc4)
