As a GCS, I tried to nursery grow commercial Poa reptans seed and could not establish a sward. The germination was extremely slow and inconsistent.
There are something like 300 identified varieties of Poa annua, so Poa greens on one course might look completely different than those on another. Poa usually begins infesting a green with coarse, bunch type, yellow-green varieties that produce an abundance of seedheads. When it’s been under a putting green maintenance regimen for several years (or decades) it tends to evolve into a finer, denser and more upright growth habit, though it may still produce a lot of seeds. The best Poa greens are normally found on old well established greens where the varieties most suited to the conditions thrive and become prominent.
Poa-free greens do exist. I managed pure bentgrass greens for three years in Tenerife in the Canary Islands, mainly due to a hot, arid climate and a slightly brackish irrigation water that did nothing to favour the Poa.