Beavers do absolutely everything humans could possibly want. Drought resistance, flood prevention, water filtration, wildfire refuge and firebreaks, riparian habitat for fish and songbirds. I'm always happy to see a story about them coming back.
First, if you have a pond beavers will come over at night and dig holes the width of a basketball about a meter from the edge of the pond into the water (vertically down, then sideways). They camouflage these holes or they simply overgrow, good luck to an unsuspecting human that had his/her leg fall into such a deep hole.
Then, they dig such holes and passages into earthworks designed as flood defenses. Huge amount of money goes into fixing such earthworks.
Finally, if you(or anyone nearby) happen to have drainage channels or small rivers, these will be blocked by beavers to the point of flooding the surrounding area. Various compensation schemes exist, but not everyone has documentation to use them.
Fibally,a braver can be a very dangerous animal when startled/provoked/cornered. Around here every person is told in their youth, "don't approach beavers" or if they bite you in the leg (usually groin area) you'll bled out in 20s.
No, beavers are not nice... If course they shouldn't be eradicated, but they should be managed properly, not enjoy 100% protection as they do here in Poland, for example.
Don't approach wild animals if you don't want them to bite you? This is just common sense.
Overall, the issues you discuss pale in comparison to the benefits they provide. Humans have built developments where naturally there would be beavers if they hadn't been wiped out. The lesson isn't that we need to fight nature harder to preserve our victories against it. The lesson is that we need to be more thoughtful so our developments aren't incompatible with beneficial natural processes.
I know how annoying it can be to have wild animals destroy your property (there are for instance beavers, racoons, wild boars and woodpeckers where I live, and I own a wooden house), but in principle we do not have more right to shaping our habitat the way we like than beavers and wildlife in general. More than "managing beavers", what we need is to find ways to co-exist peacefully. One part might be "managing" them, but another part is definitely to accept that their needs in part conflict with ours, and that we need to give them some space and accept some of the damage as a price to pay to share land with them.
Beavers can make lakes that are absolutely enormous because they aim their dams so carefully, and can construct them to very impressive heights. I've had a 20 meter wide two meter high beaver dam on my Canadian property. It seemed incredible that such tiny animals can make such enormous constructs.
Humans have done a lot by this measure. Some of it is less than desirable though.
We managed to convince the little bastards to move down the creek to a section where the flooding would hit some woods and CRP ground that we could never really farm due to terrain.
But that took like 10 years. Shooting them and destroying the dam would've taken a day.
So that's the motivation.
I don't know where you live, but, around here, humans want growth and profit. Due to this, they are not able to build a dam, but complain about floods. /s