Functions are not so rare at all, and definitely not isolated, in sequence space of biopolymers

[Rumraket (Mikkel R.)]

@Giltil

But things are very different for proteins carrying more complex functions such as, for examples, DNA polymerases or ion channels.

T-urf13 is an ion channel and evolved during selective breeding of maize.

On the evolution of Irreducible Complexity

The novel antimicrobial peptide discovered and described in Knopp et al 2019 is also an ion channel:
https://journals.asm.org/doi/full/10.1128/mBio.00837-19

Generally speaking transmembrane proteins are surprisingly easy to evolve, so much so that they are among the most likely de novo proteins.

De novo emergence of adaptive membrane proteins from thymine-rich genomic sequences

Another intriguing de novo gene evolution paper was recently published in Nature Communications: De novo emergence of adaptive membrane proteins from thymine-rich genomic sequences Some excerpts: Mutations that cause changes to the sequence or expression of established genes are typically constrained by preexisting selected effects—the specific physiological processes mediated by the gene products that are maintained by natural selection20. In contrast, emerging proto-genes are expected to mostly lack such constraints because they do not have selected effects. This would leave them more readily accessible to evolutionary changes that have the potential to increase fitness (adaptive changes)3,4. We reasoned that this initial potential for adaptive changes would give way as proto-genes mature and the adaptive changes engender novel selected effects, in turn increasing constraints and reducing the possibility of future change. This reasoning is akin to Sartre’s “existence precedes ess…

The very weak sequence constraints on transmembrane domains has also been shown phylogenetically:
https://academic.oup.com/mbe/article/33/11/2874/2272007

Transmembrane domains are so easy to evolve because they’re essentially just repeat-proteins consisting of a single, simple structural element, such as a beta-hairpin. These will naturally tend to oligomerize, their hydrophobic exterior will have an intrinsic affinity for the hydrophobic interior of the membrane bilayer, and they form so-called “barrel” (essentially just tube-shaped structures).

(http://membranproteine.net/Structure%20gallery%201.html)

There is a lot of literature on the evolution of transmembrane repeat proteins (such as beta-barrel structures).

Evolutionary pathways of repeat protein topology in bacterial outer membrane...

https://www.sciencedirect.com/science/article/pii/S0969212618302132

Etc. etc.