I know that the great Galileo made no real progress measuring the speed of light -- he disappointingly suggested that it might be infinite. I read that he concluded (based on his attempts to measure it with lanterns) that it is at least ten times the speed of sound (which he may have had a value for). Romer in 1676 brilliantly (no word play intended) measured the speed within 20 percent of the true figure observing Jovian moons and shadows at different times of the year, I think. My question is, in 1675, when Newton was over thirty, what did he think? (Or what did any other scientist of that time think?) Would he have already rejected not just infinite speed but also understand that 10 times that of sound was too low by a long shot? I have never encountered him suggested an experiment to measure the speed or an idea about this speed even though he thought about light as much as any human had.
I do not think it is a virtue to make unsupported assertions just because we happen to believe them now. Following available evidence does not makes one not smart or disappointing, that is how science should be done. Galileo deserves credit for moving the question from armchair speculations to observations and experiments, and honestly surmising what they suggested at the time. Kepler and Descartes came to the same conclusion of infinite speed around the same time based on observations of lunar eclipses and refraction experiments, see McFadden, A Brief History of the Speed of Light.
Things changed by the time of Newton. He does not refer to Romer explicitly, but asserts finite speed of light in Principia, Book I, Section XIV based on more precise observations of Jupiter's satellites (veiled reference to Romer's measurements) and Grimaldi's diffraction experiments:
"For it is now certain from the phenomena of Jupiter's Satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the earth. Moreover, the rays of light that are in our air (as lately was discovered by Grimaldus, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inflected round those bodies as if they were attracted to them..."
But Newton accepted action at a distance for gravity based on astronomical evidence, despite his metaphysical dislike of it, as did Laplace and others a century later. The speed of electricity was also widely surmised to be infinite until the 19th century, because that is what experiments done before that suggested, see Was it suspected that the speed of electricity was equal to the speed of light?