Well, you're describing the same frustration many people have downloading stuff from USGS. It can be very time consuming to download just the parts you want from their websites.
The first link you mention has a sublink to download from the national map. The sublink says you can download either as a shapefile or as a GDB, but when you drill into the data sets in the national map download page you see that some of them are only GDB, while others can be had as shapefiles or gpkg. NHDplus has only GDB enabled as far as I can see.
The national map download UI has changed since the last time I've used it, allowing you in the Advanced Search section to upload a shapefile, which I suppose could be used to limit the download to just your area of interest. (I haven't been able to get that to work, but then I haven't read their instructions.)
In the US, I usually search by state and then hover over the list of results to see the footprint of each result. I can then just download those results. When those data sets overlap as they usually do you have to mess with merging the results, though, so that can be time-consuming.
For vector layers, I usually just bite the bullet and download the whole thing. In the case of the NHDplus data set you're lucky USGS is willing to give you a zip file that has the whole thing, a mere 26.2 GB. Download the whole thing, and then some weekend unzip it, import into Manifold, save as a .map, and then start carving out only the portions you want. It will be slow at first but because the intermediate steps (download, unzip, import, save as .map, clip...) you can start and then forget about overnight or whatever, it might be easier than spending lots of interactive time trying to figure out how to download the parts you really want and then merging them where they overlap.
As for slow servers, that's part of the deal with USGS. Unless you want really tiny sections of data, they are usually far slower than just downloading the whole thing and cutting out what parts you want using Release 9.