I'm trying to check the following system for for time-invariance.
$$ y(t) = \int_0^t x(\lambda) d\lambda$$
Please explain me why it is time-invariant.
Note that there are different kinds of (causal) integrators. The one in your question with a fixed finite lower integration limit is not time-invariant, as you've shown yourself (in the comments).
Other variants are
Both of these integrators are time-invariant, which is straightforward to show.