To answer the first phrasing of the question: "Not yet". Approximate answer to the second phrasing below, after discussion. Adding to GrapefruitIsAwesome's answer, there are a couple of problems in addition to Daylight and Cloud Cover:
Once every 28 days, the moon is full and makes the whole sky glow (in the same way that the sun does during the day, but to a much lesser extent). As this makes it impossible to see faint objects, all surveys shut down for 3-7 days for this period (depending on the survey and what they consider to be faint), leaving the sky more or less unmonitored. Let's say this means we only see around 85% of what we would otherwise see.
For non luminous objects, the amount of light received from them by an Earth based optical telescope is heavily influenced by their solar elongation (angle between the Sun, the object and the observer). I.e. even if it is night and we are looking at the right part of the sky as an asteroid whizzes by, we get to see very little of the sunlight reflecting off it. This means there is in fact a very limited part of the sky where we manage to observe non-luminous objects. Because of this, over half of the asteroid discoveries we make are in a narrow cone facing directly away from the sun, which covers less than 4% of the sky. See Wikipedia. Let's say this means we only see around 8% of non luminous objects we would otherwise see.
In fact these issues are related. The moon is so bright when it is full because it is in opposition, and significantly dimmer the rest of the time. However, the Sun is much brighter than the moon, and never dims, so Daylight is definitely a bigger problem. It's easy to forget how big an issue this is.
Because we see the day/night cycle from the spinning earth it's easy to think "I only need to wait a few hours to see that bit of the sky". Unfortunately of course it's you that's moved. The daylight stays in roughly the same direction in the sky. To see that part of the universe you need to wait months, not hours for it to be night... and the bit of space between Earth and the sun, you never get to see with an ordinary telescope*. Let's say this means we only see around 50% of what we would otherwise see.
So in an attempt to give an approximate answer to "how likely would we be to notice it":
- For luminous objects: P(not full moon) 0.85 x P(not in daylight) 0.5 = ~43%
- For non luminous objects: P(not full moon) 0.85 x P(illuminated enough) 0.08 = ~4%
Note these approximate numbers ignore cloud cover, airglow and various other lesser factors, and additionally limited by the sensitivity of current telescopes.
* For this reason there's a space telescope being built (NEOSM) that will orbit at Sun-Earth L1, the point at which Earth and the Sun's gravity cancel out. It will look back towards Earth with the Sun behind it, so that we can at least see into the part of space between it's orbit and the Earth. For reference it's about 1% closer to the sun than we are, which may not sound like much, but that's still about 4 times further than the moon. That's the LaGrange point on the opposite side of Earth to the one that Webb is currently orbiting, L2. Both are around 1.5 million km from Earth