Most of the times when a big club plays a minnow in the earlier rounds the games aren't even on television. The networks over there like to make money just like the ones over here do, and they know that showing a game between two Premier League teams will draw far more viewers than watching Man City's scrubs pummel some League One team.
The first round of games that Premier League teams play in is round three. From January's round three games, the BBC, which is the only free coverage of the FA Cup in England, showed Liverpool versus Everton and Arsenal versus Leads. The Arsenal - Leeds game was a Monday game and the only game played that day, so it's easy to see why that was picked. Also note that Leeds isn't a minnow, they are the league leaders in the Championship. Notice the other game they showed was one featuring two EPL teams.
Then there are also games shown on a pay channel, BT Sports. They showed Newcastle at Rochdale and Man United at Wolves (note, an all EPL matchup) on Saturday and on Sunday they showed Middlesbrough (a team that has frequently been in the Premier League) versus Tottenham and West Ham versus Gillingham.
So that's six games and only two of them feature teams that would be considered minnows. And neither of those two minnows played one of the big clubs. Note that they could have shown Man City - Port Vale, the kind of game that you think that they love, and they did not. Note that they could have shown Chelsea - Nottingham Forest, the kind of game that you think that they love, and they did not. And the reason for that, of course, is that they don't actually love showing those games, and the reason that they don't is because people don't watch them.