Note that with the magic of exponential growth, a few probes sent long distances and then spreading out locally should be able to reach every star in a galaxy with surprisingly little extra time over simply making the trip.
Even hand waving that we don’t have force fields to survive single gram impacts at 1% light speed. At those velocities anything is going to act like a massive shaped charge. Space is fairly empty, but when you start talking about the volume swept up across light years that’s really not a safe assumption.
70 years ago people generally assumed fusion was right around the corner. It’s only when you start building these things you get some idea of all the real limitations.
Now it is absolutely true that we could not survive any significant impacts at that speed. However we also have data indicating that the interstellar medium is essentially empty.
Here is a sanity check. Per http://cdms.berkeley.edu/Education/DMpages/FAQ/question36.ht... the density of dark matter (that is, all the stuff that we can't see) is around 6x10^(-28) kg/cm^3. Since you arbitrarily said a gram, let's switch that to g/m^3, and we get 6x10^(-19) g/m^3. The distance to Alpha Centauri is on the order of 4x10^16 m. So a 1000 square meter surface should run into about a gram of "stuff". According to our best theories, most of it is currently unknown stuff, 99% of the rest is single atoms, and the vast majority of the rest is dust particles with sizes on the scale of waves of blue light.
Dealing with that dust and hydrogen is a serious issue. But the odds of running into pebble sized objects at random are pretty low. Heck, even microbe sized objects will be vanishingly rare. True, it will be a disaster if we do. But that translates in practice into, "There will be a failure rate for probes."
The size of a probe to bootstrap manufacturing seems likely to be a lot smaller than a billion tons of cargo. Don't forget, this is going to be encoded using technology that is as far beyond us as we are beyond building the pyramids. I think we can assume rather efficient storage/bootstrapping mechanisms to be available for us.
As to your sanity check, Alpha Centauri our closest star, if we are sending multiple probes then you need to targets much further out. Especially if we are trying to get somewhere approaching a straight line in 3D space. The 20th closest star system is 12 light years from earth and on the order of distances we need. Though this gets worse if you need specific kinds of star systems.
That’s pushing 3+ grams under your estimate and your 30 ton payload would need truly massive shielding. Or, you can have a 30 ton shield being activly maintained in front of a postage stamp sized cargo and possibly survive the trip.
As to that billion tons of cargo, it’s almost unimaginably better than current tech could do. We are talking either self sustaining habitats for hundreds of people or safe replicating near full AI on million year timescales. It’s got to gather energy and resources, build CPU’s, engines, etc to gather even more resources. It’s got to travel hundreds of years meaning redundancy and self repair on all critical systems etc etc the list just keeps going.
In terms of engendering from ancient Egypt to now the tallest structure is just under 6x as tall. Some things have improved vastly more and others even less. We simply can’t assume magic even on 10,000 year time scales.
The Milky Way is 105,700 light years across. At voyager probe speeds that’s ~1.8 billion years. Stopping requires 4x as much energy, and cutting that time in half requires 4x as much energy. Worse slowing down to build a probe adds more time as does however long it takes to select targets, gather resources, and build several more probes.
On top of a huge number of issues could slow things down such as lost probes, or needing to avoid the center of the galaxy resulting in an even longer trip. Not to mention collisions, at a minimum you need to survive grain of sand sized collisions, but interstellar space may contain significant numbers of much larger objects.
Finally these probes would both come here, and stick around for us to see. A true space based civilization might have little interest in planets like ours, or have sent multiple probes to our solar system that failed.
It seems possible for hundreds of civilizations to all be expanding across the galaxy even if we have seen none of them.