While this is an exceedingly broad question, I'll mention a few relevant things.

I am interested in learning more because for many cryptographic applications, we pick basis from these families in which to solve the "closest lattice point" problem, and depending on which basis is chosen can have dramatic implications on the complexity of the solution.

In standard lattice-based cryptography this isn't really true.
Essentially every lattice-based hardness assumption I am aware of admits what are known as *randomized self-reductions*.
At a high level, these say that worst-case complexity of a problem and the average-case complexity of the problem are roughly the same.
Not all hardness assumptions admit such reductions --- the worst-case complexity of factoring is vastly different than the average-case complexity, which is why RSA has to be defined relative to semi-primes rather than arbitrary integers.

Moreover, most lattice-based hardness assumptions are not actually that straightforward to write down in this "good basis bad basis" paradigm.
For example, (one form of) the most popular lattice-based hardness assumption is

**Learning with Errors Problem**: Let $n,q, B\in\mathbb{N}$. Let $A\gets \mathsf{Unif}(\mathbb{Z}_q^{n\times n})$, $\vec s\gets \mathsf{Unif}(\mathbb{Z}_q^n)$, and $e\gets [-B, B]^n\cap\mathbb{Z}^n$.
The task is then to distinguish between samples of the form
$$(A, A\vec s + \vec e)$$
and
$$(A, \vec u)$$
where $\vec u\gets \mathsf{Unif}(\mathbb{Z}_q^n)$ is uniformly random.

To phrase this in terms of lattices, define the lattice $\Lambda_q(A) := \{y\in\mathbb{Z}^n\mid \exists x\in\mathbb{Z}^n\text{ s.t. }y \equiv Ax\bmod q\}$.
Note that this is a quite natural object --- you get it by applying the "Construction A" to a uniformly random $q$-ary code.
In terms of this lattice, the (decision) LWE problem asks you to distinguish between

- uniformly random elements of space, and
- points that are unnaturally close to this lattice.

The search version of the LWE problem is to recover $\vec s$ from $(A, A\vec s + \vec e)$.
This is equivalent to solving a variant of the closest vector problem on this lattice (it is a "promise" variant of the problem).

This all being said, there is not an obvious good basis/bad basis perspective LWE-based encryption.
One can phrase things in terms of knowledge of a specific short dual vector related to $\vec s$ (which often can be used as a terms of "good basis"), but this isn't the most natural presentation of things in my opinion, and is not the most common perspective taken in the literature.

What you are talking about is instead most similar to

- encryption based on the NTRU assumption,
- how lattice-based signatures are constructed, as well as
- the recently-introduced lattice isomorphism problem (LIP).

Of the three, the last matches your idea of being "stuck within a family, necessitating picking a good family" the best (the others just "pick a family randomly" essentially).
There are a few papers about cryptography from LIP, but it's really quite a new assumption (I think two years old at the oldest --- the problem itself has existed for longer, but cryptographic constructions are recent).
Relevant pointers are Bennett et al '21, Ducas and van Woerden, and papers that cite these papers.
Of the two Ducas and van Woerden matches your perspective the most closely, so is perhaps the best to look into.