Our dollar coins aren't that heavy... certainly lighter than $0.50 or $0.20's (we dont do 'quarters' here...).
We ditched $1 and $2 probably 20 years back.. and yes it can be annoying when you have $50 worth of the buggers.. but I just make sure to 'reuse' or make 'exact change' with them when out and the rest goes into my "Camera Slush Fund Jar" - so its win-win... the money is removed from the houses "Consolidated Revenue" and I get to buy stuff every couple of months...
Out of curiosity - why dont the US switch to Polymer notes ? Less able to be counterfit (more security measures possible etc), and heaps more durable. The 'cost' of low-demonination paper money is that its usually high-circulation, and thus, degrades quicker and must be destroyed as its no longer legible enough to be considered 'legal tender'. They then have to produce a new note, as well as have the old ones removed from circulation and destroyed.
Switching to Polymer, you can keep your $1 bills, and reduce costs because they last 4-5 times longer than paper money in equivalent circulation.
Polymer banknote - Wikipedia, the free encyclopedia