The Premier League winner gets to claim to be the best team in the country, that's the league. The FA cup winner doesn't get that privilege really because the FA Cup has less prestige than the Premier League title. That goes for all domestic league and cup competitions.
Anyway, playing against teams from higher leagues is certainly exciting for the lower league teams and fans, but that isn't really the case for the top teams in the Premier League. The exciting part for them is playing against top teams in a knockout game.
That's not how it works, but go off