Why Lie algebras if what we care about in physics are groups?
The objects which matter in physics are Lie groups and not Lie algebras. Lie algebras approximate only infinitesimal group transformations and in quantum mechanics the finite and global properties of the transformations matter.
However, (considering quantum systems with a finite number of degrees of freedom), the spaces of quantum states are projective, as there is no physical meaning to the overall magnitudes and global phases of state vectors. Thus, symmetry groups act on the spaces of states via projective representations.
For a semisimple compact Lie group, a projective representation is a true representation of its (simply connected) universal covering. The representations of the universal covering group are in a $1-1$ correspondence with the representations of its Lie algebra (which is the same Lie algebra as the original group) . This is the reason why all representations of the group's Lie algebra can appear as realizations of symmetries in quantum systems.
May be the most famous case is the rotation group $SO(3)$, which can be parametrized by The Euler's angles. The true representations of the rotation groups are the integer spin representations. However, there are quantum systems in which the rotation symmetry is realized by means of the half-integer spin representation (such as the electron spin or a qubit). The half integer representations are only projective representations of the rotation group; however, they are true representations of its universal covering $SU(2)$. The representations of $SU(2)$ are in a $1-1$ correspondence to the representations of the isomorphic Lie algebras of both groups $\mathfrak{so}(3) \cong \mathfrak{su}(2)$.
It a general fact that any Lie group representation induces a corresponding Lie algebra representation (but not necessarily the other way around). Therefore, short of topological informations, we can often learn a great deal about physics by dealing with the Lie algebra (which in practice is a mathematically easier gadget to handle, namely just a vector space).
Of course, a full treatment would include investigating whether the Lie algebra representations can be lifted to consistent (possible projective$^1$) Lie group representations of the theory. This analysis is often glossed over in physics textbooks.
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$^1$ Whether projective representations are allowed depends on context.
Observables often close on a Lie algebra, and their matrix elements are directly related to measurable quantities, v.g. average values, eigenvalues (spectrum), transition rates to name a few.
In addition, there are some very useful algebras for which there is no group, v.g. the Temperley-Lieb algrebra, $q$-deformations, etc.
Finally, the equations of physics tend to be expressed in differential form, comparing functions that are infinitesimally close. Thus it’s no surprise that infinitesimal generators, i.e. elements of the algebra, tend to be more useful than group elements, which require (sometimes very intricate) exponentiation.