No, the intuition doesn't really scale very well, especially not if you want the optimal solution.

The problem is known as Independent Set and is not only NP-complete, but even W[1]-complete.

There are heuristics you can use, e.g., pick the smallest degree neighbor into your solution, delete the neighborhood, and rinse and repeat. Of course this is not guaranteed to give a good answer.

Another way is to use LP (rounding) for finding the smallest vertex cover. The complement of a vertex cover is an independent set.

No, the intuition doesn't really scale very well, especially not if you want the optimal solution.

The problem is known as Independent Set and is not only NP-complete, but even W[1]-complete.

You can try the following greedy algorithm: pick the smallest degree neighbor into your solution, delete the neighborhood, and rinse and repeat. Of course this is not guaranteed to give a very good answer, but it does give a solution which is not more than a factor $1/\Delta$ smaller than the optimal, where $\Delta$ is the maximum degree of the input graph.

Another alternative is to use LP rounding for finding the smallest vertex cover. The complement of a vertex cover is an independent set. You can also experiment with using randomized rounding:

LP relaxation for I.S.: maximize $\sum_{v \in V} x_v$ constrained to $x_u + x_v \leq 1$ for every edge $uv \in E$, and $0 \leq x_v \leq 1$ for all $v \in V$. With randomized rounding, we take the variable $x_v$ as the probability of selecting $v$ to the solution. Finally clean up the solution by removing arbitrarily endpoints of edges in the solution.