The very few attacks/half chances Burnley had/created, set pieces included, they looked like scoring from. It's not some crazy agenda against Liverpool's defence, it's reality. Taking risks up the pitch doesn't stop you from defending set pieces well, or being organised enough to deal with a long direct ball. These are the basics. It's not like your CB's were exposed in some break away 3v2/4v2 situation for these chances.
For the goal, MOTD says Robertson should've been more central, I'm not buying that tbh. 2 CB's go for the same ball (and got beat to it), in fairness to Matip, he's in a decent position, could've been more aggressive perhaps, but in the right sort of area. But what Klavan does here I'll never know?
^ Clearly sees the situation, all he needs to is hold his position more or less and cover Matip, not 'charge' into his battle.
I use 'charge' very loosely, as look at his flimsy 'ah don't hurt me' body language when he eventually gets there:
Completely vacates the space. Had he of been in this area, which he should've, there's a strong possibility he blocks the shot, or maybe even stop the ball getting to Arfield.
Definitely communication/concentration issues, which again, tends to happen in these profile of games in which the opposition are there to absorb pressure, ride their luck, and try to completely sucker punch you in targeting identified weaknesses.
Should you have scored more than 1 goal? Absolutely. After last season, nobody knows this better than United fans. But should you be able to deal with the 2 or 3 threats posed from negative opposition? 100%. As much as you don't like people saying it, you struggle to do so. Attacks win games, defences win championships.
Maybe you do have systematic flaws defensively, but you can't deny a more competent CB deals with this situation better than Klavan.