To back up your first point, here's the list of teams to have won each major competition in England in the last 10 years.
Premier League
Last 10 years: 4 clubs - Manchester City 6, Chelsea 2, Leicester City 1, Liverpool 1
Last 15 years: 5 clubs - Manchester City 8, Chelsea 3, Manchester United 2, Leicester City 1, Liverpool 1
FA Cup
Last 10 years: 5 clubs - Arsenal 3, Manchester City 2, Manchester United 2, Chelsea 1, Liverpool 1, Leicester City 1
Last 15 years: 6 clubs - Arsenal 4, Manchester City 3, Chelsea 3, Manchester United 2, Liverpool 1, Leicester City 1, Wigan Athletic 1
EFL Cup
Last 10 years: 4 clubs - Manchester City 5, Manchester United 2, Liverpool 2, Chelsea 1
Last 15 years: 6 clubs - Manchester City 6, Manchester United 3, Liverpool 3, Chelsea 1, Swansea City 1, Birmingham City 1
That means that 6 clubs have shared the last 30 major trophies. Even if the timeframe is extended back 5 years only Swansea, Birmingham (both EFL cup) and Wigan (FA Cup) get added: that's 9 clubs winning 45 trophies and of those 9, 4 have a total of five wins leaving 5 clubs sharing the other 40 competitions.
Last 30 titles: Manchester City 13, Chelsea 4, Liverpool 4, Manchester United 4, Arsenal 3, Leicester City 2
Last 45 titles: Manchester City 17, Chelsea 7, Manchester United 7, Liverpool 5, Arsenal 4, Leicester City 2, Birmingham City 1, Swansea City 1, Wigan Athletic 1