The EU might have retained popular respect with UK voters if the MEPs that the UK sent to parliament had voted in accordance with their electorate's desires rather than not-voted-because-they-are-a-protest.
Or not. Quite a lot of stuff in the media that is anti-EU, too. That might be more powerful.
I'm not sure what you mean by a tiny minority. IIRC the UK Parliament voted against Maastricht twice and it went through on the third attempt. I think UKIP has exploited and perhaps stoked the skepticism, but IMO did not create it.
As for polling, it looks like my memory doesn't quite match reality: http://theconversation.com/polling-history-40-years-of-briti...
This polling history shows wild oscillation over time, and the EU was formed from its predecessor just as things were close. I had been under the impression it was a gradual shift from the high of 1979 to the low of 2016.
Regardless, "stoked" sounds like a good description of UKIP's behaviour to me, so I think we may be agreeing with each other.
I think the Leavers were dissatisfied from the beginning. You can directly trace the formation of UKIP from the Maastricht Rebels. https://en.wikipedia.org/wiki/Maastricht_Rebels in the Major government.
UKIP was formed out of protest at what they saw as the EU going off in a highly political direction (i.e. becoming a lot more than just an economic area).
They may be wrong but the objections are based on fundamentals.