Sounds like we're going backwards! Soon we'll be collaborating with other nations about other things too!! /s
-wait why are you running away? This is exactly the same thing as a trade deal!
The country with the entirely unelected unremovable upper house has no business complaining about this.
Besides, short of violent revolution there's nothing the UK population can immediately do about the HoL. The Brexit vote was something that the people could directly do to at least de-abstract power and bring it closer to home. This was one of the most-advertised perks of the EU by the remain campaign: It's not the same as a nation superstate because you're able to voluntarily leave. Brexit was just the UK population exercising that right.
The other point is that the Lords really doesn't have that much power when all's said and done. Yes, they can quibble and send bills back for review a few times, but they're nothing like the US senate for example, the commons have the final say over everyone else. Compare to the EU commission, who have much more power, and actually make the laws.
But only in a few domains, and they always require approval. From wiki: "Most legislation needs to be proposed by the European Commission and approved by the Council of the European Union and European Parliament to become law." The former consists of the national governments, which usually has a popular mandate, the latter is chosen directly.
Both the UK and the EU are only nominally democratic.
Substantial political union isn't required per se, just enough common interest. That at least was the original thought behind the predecessor of the EU, and it seems to have worked through sharing economic interests.
You can take any trade bloc out there - none has the influence or the prospects of the EU, they are all fundamentally weak. You need shared political force to make a difference.