Doesn't modern electronics require massive shielding?

e.g. you would need too much shielding and would go over the 1 gram restriction.

If you have enough of them you can just lose some fraction and still function.

Also it may work out that a lighter, faster probe with a shorter service life could have a greater effective range, if the service life is less sensitive to weight changes than the speed is

Right, that’s a nice way of thinking about it.

And how do you get the data back from these probes? The voyagers have antennas that are close to 4 meters in diameter and ~25-watt radio transmitters. You aren't doing that in a gram, and you aren't powering that in a gram either.

Just go read the starshot proposal. They address all of these issues, some more convincingly than others but they have thought it out in great detail.

In short, it’s a swarm of gram-scale probes and they work together to transmit.