Occam's razor tells me: Its easy for the local politicians to blame "Brussels" for any problem they dont want to deal with / take responsibility for. Its also easy for the population of the respective country to accept this framing, since that relieves them of any responsibility of having voted "the wrong people" into power, or not engaging in politics themselves.
Regarding the "unelected" European Commission: Its members are proposed by the European Council(which is composed of the elected governments of the EU members states) and then subject to a vote of approval by the elected European Parliament. I find it hard to believe that your opinion of the European Commission (or the EU in general) would differ significantly if the European Commission would be elected directly.