Is the US short on politicians given the spread of power? The US has thousands of State Legislators and a bit more power than Australian states for example. In addition you have Counties and Municipalities which when compared to Australia take on many of the roles we give to state governments such as police and schools.
So while the US doesn't have a lot of representatives at the Federal level there are a lot of representatives at lower levels of Government who generate considerable tax revenue and exercise considerable power and influence discharging their duties. Maybe that is enough?
Sadly, it isn't. Local elections are far too often ignored. Earlier this year, I tried to be placed on the ballot for the Tennessee State House. All I needed was 25-30 valid signatures of residents in my county and the county next over, and I would be on the ballot against the local Republican politician. I didn't get that far thanks to my own laziness (I was actually noted in the county paper as one of the candidates who applied but wasn't placed on the ballot). That Republican is now running unopposed. It seems that in elections where the Republican runs against either a third party or a random non-established independent, the Republican wins, but the other candidate still gets a significant amount of votes. And I'd be nothing short of amused if thousands of people were voting for me, even if I handedly lost.
These low barriers to election for state legislatures are very common in the US. And most Americans tend to ignore their state legislature and focus way too much on the politicians in DC. It's true that those in their state have way more influence on them, but not a lot of people in the US know that, hence why that state elections often have horrible turnout, especially in non-presidential election years.
The counties and municipalities, well, most all of them are commonly perceived as corrupt by locals. I know my own county (Cheatham County, Tennessee) has had corruption issues in the recent past, in the county supervisor position, several mayors of towns, and especially the sheriff's department. By that, I would suspect that most American's trust in their local government isn't very high, be they living in a rural area or urban area. People seem to much more trust their politician in DC than the politicians in the state, many of whom they might not even know the name of. I suspect that trust in the national government in DC is higher than both state and local governance. These are just common analyses based on what I've encountered. I also know that these local politicians tend to be both approachable and yet, well, your typical "politician" type. Case in point, a certain Tennessee State Representative
Tommy Head represented my district when I was growing up, and he had numerous encounters with my father (either in my father's job--he needed construction equipment, my father dealt in that field--or once on a golf course of all places), and each time when my father questioned him on certain votes and certain positions he had taken, Rep. Head evidently displayed remarkable ways of avoiding the question and presenting himself as "neutral" by resorting to "well, my constituents say..." as a counter-argument.
All of this, I think, is a very good illustration on local government in the United States and the common perception of it. I notice, though, when you make it a national matter, behaviour like the aforementioned Tommy Head's becomes less common and less known. It becomes an American matter, and since less people know their senator or representative, you don't hear as much bad things and less stories about them are spread. That's basically why a larger House (maybe 700-800 in terms of size) could only be for the benefit of American democracy. The Wyoming Rule, of course, is only a start.
How do you convince people of this? Well...that's gonna be a problem. But voters could certainly be convinced.