Democracy is from the Greek “demos” (people) “kraita” (power/rule).
A referendum is a direct vote in which an entire electorate is invited to vote on a particular proposal.
So your position that a referendum “is not democratic” is unsupported. It meets the definition of democracy. As with everything in life, it is flawed. But that does not make it “inherently anti-democratic.”
And in the case of Brexit, the referendum was used as a democratic device of last resort. So even if you agree that referendums are not ideal, this was the only remaining democratic device left to try to resolve the issue.
On 9th June 2015, when other means to reach consensus on the EU had failed, our democratically elected representatives voted to hold a referendum. 544 to 53.
The people we elect to make decisions on our behalf voted amongst themselves to delegate the decision for Brexit to the people. Again, this is hardly undemocratic.
You mention inclusive processes and compromise. Brexit was the endpoint of a multi-decadal failure to compromise. For example: UK/EU migration policy moved against popular opinion for nearly two decades. You are entitled to disagree with the popular opinion, but if you are as concerned about democracy as your question implies you will quickly see my point here.
At any point before Brexit policy (in any number of areas, not just in migration) could have been changed to better represent the views of the people, but it wasn’t. Partly this is due to the design of the EU, but there are other reasons too outside the scope of this answer.
Finally, enough democratic pressure was exerted via UKIP winning the European Elections in 2014 such that a referendum had to be offered by the Conservatives to avoid losing the democratic General Election in 2015.
This would be the first time the UK had ever had a direct say over UK membership of the EU since its creation in 1992. The conversion of the EC to the EU was not put to a vote in the UK. It could easily be argued that if there had been a failure of democracy it happened well before 2016.
Furthermore, on some issues, at their base, there is no compromise. You either have nuclear weapons, or you do not. You either have gay marriage or you do not. You are either a member of the EU, or you are not. So the lack of compromise in the 2016 vote is a characteristic of the burning question, and not indicative of a lack of democracy: Leave or Remain? Pick one.
And these problems do occur with “ordinary political processes” (your phrase). I might disagree with numerous policies the Government has in place. My view is not represented in those areas. Just like your view is not represented to your satisfaction in this area.
As to the benefits of referenda? They are simple, clear and unambiguous. The result is about as unarguable as we ever get if we believe in “people power” (aka democracy, aka the wisdom of crowds). But they can be disruptive. So they are typically reserved for big questions.
So in answer to your question: I reject the premise. Referenda are not anti-democratic. And Brexit wasn’t a failure of democracy: on the contrary it was democracy winning out after elected representatives had studiously ignored and neglected popular opinion since 1992.