This is something I know a fair bit about. If you don't build in areas susceptible to flooding in the first place (including but not limited to wetlands), you don't need to engineer around nature. And it turns out that our engineering design floods are not keeping up with real floods, so all that engineering is not really helping.

Plus, engineered structures need regular maintenance and we are generally not all that great at infrastructure maintenance. See the Libya damn disaster for just the mort recent example.

Which one's more sustainable and budget friendly?

We can't beat nature on this one. All our attempts to prevent flooding are expensive and just move the issue elsewhere. Working with nature is the way to go.

Dams can fail. Just look at Libia right now. And I know not every country is Libia, but pretty much all dams in the US are fucked too and nothing is being done about it. This is all very expensive infrastructure that needs constant upkeep and investment and is disastrous if that for some reason isn't being done. Here in the Netherlands, there has been a massive project the past like 20 years to 'give rivers space' with wetlands and floodplains. It works well. I mean, we are literally the best in the world in managing water. 2 years ago a couple of rivers flooded. The city I lived in was already done with their part of the project. The floodplains flooded and everyone was safe, but another city which wasn't done yet flooded and some cities in Germany and Belgium did too.

Wetlands managed by beaver dams do this stuff but also recharge aquifers, promote formation of peat lands holding carbon under water and eventually in the soil, change local climate increasing rainfall and sustain forests and ecosystems that have further effects to enrich soil and keep areas livable for many centuries and millennia. This form of land management fights back desertification leaving lands better than they were and prevents land subsidence in coastal estuaries. Draining swamps with systems of canals, levees and dams for human habitation and exploitation, degrades soil, aquifers and biodiversity leading to deforestation, desertification, and places like the Louisiana estuary being slowly lost to the sea in the span of just generations.

Wetlands and floodplains are very important to mitigating floods, soil retention, and mass filtering. We haven't beaten nature and are spending significant sums of money to revitalize and protect these areas because we have realized that they are just better than what we do.

It's one, but you aren't supposed to build that close to the wetland. Like a drain in a house, just because, you should always want water to go somewhere if its on the floor else would have long term problems. Part of the reason street roads are not flat.

The image is the equivalent of choosing to build a house on low tide. and wouldn't realistically happen.

I've got some prime real estate in New Orleans to sell you.

There are a couple of great "practical engineering" videos on the topic of you're interested:

https://youtube.com/watch?v=UBivwxBgdPQ&si=4gPaAIjc_Lpr9mXh

https://youtube.com/watch?v=vLZElIYHmAI&si=LN61GGBKK88oImkF