It is perfectly reasonable to infer a link between the murder of Jo Cox by a deranged fanatic, identifying himself as a "patriot" and the increasingly xenophobic rhetoric of elements in the leave campaign.

I would suspect that there will be a good number of till now undecided voters who will come down on the Remain side because they infer such a link.

That is not the same thing as suggesting a direct link. And so far as I'm aware no one has suggested or implied any such link.
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As has been pointed out, government ministers are not elected any more than EU commissioners. The process is that the Commission President, who is elected by the European Parliament for a term of five years, is presented with a list of commissioners appointed by EU govenments, and chooses which posts to give them. The complete list of appointed commissioners is then taken back, and is accepted or rejected a whole.

In a sense this is fairly similar to the Westminster model, where Parliament approves a Prime Minister who has been appointed formally by the reigning monarch, on the basis that he or she is able to win a vote of confidence in the Commons. The Prime Minister has absolute authority to appoint anyone he or she likes to be a Minister (they can always be appointed to the Alords if need be), and the Commns has no say in it.

Both systems are partly democratic, partly the reverse. The EU model, constrained by the non-federal structure of the EU, means that by definition Commissioners cannot all come from the same party, and that the whole system is intrinsically multi-party. This has disadvanrages as well as advantages. But not many Brexiters complaining about a democratic shhortfall are advocating a move towards a more fedderal system.

It is possible to think of reforms that would increase democratic control, but of course tyat can only be brought about by people in the EU acting together.