If a working telescope was barely possible at the time of Digges (circa 1570s), what technical advances allowed Lippershey (patent 1608) and especially Galileo (1609 onwards) to create working telescopes? In Galileo's case, he could easily view Jupiter's moons, craters on the Moon, and probably even Neptune (Jupiter aligned with Neptune and Galileo appears to have dutifully recorded it as a fifth moon), so his telescope was "pretty good" for the era.
One of the main technological advancements that allowed Lippershey and Galileo to make their comparatively powerful telescopes was a change in how lenses were manufactured.
Somewhat sadly, the exact date and progenitor of this new innovative method is not clearly known, as according to the Galileo Project's telescope webpage:
the telescope was not the invention of scientists; rather, it was the product of craftsmen. For that reason, much of its origin is inaccessible to us since craftsmen were by and large illiterate and therefore historically often invisible.
The change, according to the American Society of Mechanical Engineers' webpage Only as Good as Their Telescopes, up to a certain point in the late 16th century
Shaping lenses involved increasingly sophisticated use of primitive tools, all of which were some variation on the basic lathe. The operator tried to form a lens contour by comparing it to a metal gauge that had an edge cut to match a compass-scribed arc.
The change came about when this practice was replaced with a process that used
a series of turning operations in which the tools, rather than the lens itself, were lathe-turned. Lenses were produced by grinding them against a metal tool called a "lap," which had been produced using a lathe.
According to the Galileo Project Lippershey webpage, these new lens crafting techniques were introduced to the Dutch (where Lippershey was) from the Italians in the 1590's.
One name does stand out, Ippolito Francini (1593-1653), was a lens maker who was one of the pioneers of this new method, in collaboration with Galileo, he participated
in the construction of lenses for telescopes, achieving substantial fame. He is credited with introducing the vertical optical lathe, a design still in use.