Commissioners are not voted for by the general public directly. Each country appoints one. Why is the general public not annoyed at the undemocratic Prime Minister ? He's not voted for directly either. Same for each member of the cabinet.
As for the countries appointing a Commissioner, it doesn't matter as Commissioners have a clear mandate to avoid taking instructions from their own government. In other words, they don't represent the people that appoint them either. If you don't believe me, watch this video from the European Commission's own YouTube channel, see if you can spot the bit about not taking instructions from their own countries...
The upcoming Conservative leader elections will elect the new prime minister, and will not be a direct democratic election of the British populace, so that would seem to support the upthread claim that the PM is not directly, democratically elected.
In principal, as people are voting for the best combination of MP candidate and party that suits them, there's no need for a new election to decide the new prime minister as the Conservatives still hold the parliamentary majority. In practice, as most people know very little about their MPs and are voting based on the personality and perceived competency of the party leader (and/or identification with the principals of the party), there will be people calling for a new general election.
I didn't vote for the Conservatives, and I'd like to see Labour take their place, but now is not the right time to do so. Not only have we got the uncertainty surrounding Brexit, there's also a civil war going on in the Labour party, which is likely to get worse when the report from the Chilcot Inquiry on the Iraq war is released next week. We need stability right now, new UK elections can wait until 2020.
Also, the leader of each party is voted for by the members of that party, and anyone can become a member of a party by paying a small membership fee. If you care about who is the leader of a particular party, you do have the choice to have a say in it.