No, that's not it. The EU has a reputation for red tape and bureaucracy within Europe too, that's not a UK thing. It's a running gag in the game Tropico for example, which is made by a German company.

And there are good reasons for that. The EU likes to "reduce" red tape by replacing country-specific systems with its own. This can be seen as a reduction if you believe everyone in Europe had to handle every other countries systems simultaneously, and there are (mostly business) cases where that was sometimes actually true. But the vast majority of the time it wasn't. Most people, most of the time, were handling only their own countries bureaucracy and trading with firms in their own country, or one or two others at most.

If your country has a relatively lightweight process which is tailored for local needs and well understood by local people, which is then "harmonized" by the EU, it's both possible and common that the amount of red tape in your life increases. The fact that there are now fewer systems across Europe than before doesn't make any difference because you weren't interacting with them anyway.

In the UK there are several reasons why the EU got a really bad reputation for regulation:

1. Many EU regulations are newly created and regulate aspects of daily life where there is no obvious trade benefit and nobody was asking for it. Example: around the time of the referendum the Commission was trying to introduce regulations that would "harmonize" the power of kettles, which would mean they'd take far longer to boil than before. Guess which country boils kettles way more than any other? They quietly delayed the introduction of these rules, because they knew it'd piss off Brits and make them more likely to vote Leave.

2. There have been many cases where something was happening that both voters and politicians thought was obviously stupid, yet, politicians couldn't fix it because EU rules prevented it. An example of this is the deportation of terrorists, which EU rules routinely blocked on grounds many voters thought weren't very good (e.g. right to family life). Again this has nothing to do with reducing bureaucracy or making trade easier, it's just about controlling local decisions.

3. Many EU regulations that annoy people on a daily basis, e.g. cookie banners and voters never cared about any of this. Their priorities have never been internet related things.

4. From the British viewpoint EU rules are often seen as vague and arbitrarily or unfairly enforced. There are many examples that could be used to illustrate that.

Since Brexit there have been a steady stream of situations that fit this template: media/voters get upset about government doing something that looks stupid, government comes under pressure and fixes it, then says "Oh and actually, it turns out the ability to do this is another Brexit freedom that wouldn't have been possible in the EU". Invariably whatever they're doing seems obvious and nothing special, without any obvious reason why EU bureaucracy would have stopped it. This reinforces the impression that there must have been far more EU rules on stuff than anyone realized. It raises the obvious question - how many obvious things weren't getting done for decades, because they violated some rule and so politicians or bureaucrats just ignored them completely? Remember, the iron law of the EU is nobody can fix anything.