When using a solar filter and properly exposing to see details on the sun's surface, the only thing that is going to show up anywhere in the frame is the sun. Notice that there are no stars visible in any filtered images below.

About an hour before totality as the Moon's silhouette is just beginning to move in front of the Sun. A solar filter was used with a 200mm lens (the image is heavily cropped to about a 900mm angle of view on a FF camera) on a 20MP APS-C camera. ISO 400, 1/500 second, f/5.6. The filter is providing about 15 stops of attenuation not only of visible light, but of infrared and UV as well.

Everything else will be entirely too dark. If you want to take photos of the landscape during an eclipse, you won't be able to use a solar filter unless you use exposures in the minutes-long range.

This unfiltered image was taken about two minutes before totality. ISO 1600, 1/100, f/4 That's EV100 = 7, about the same brightness as brightly lit nighttime streets or a stage show. In full direct sunlight the scene would have been about eight stops, or 256X brighter!

During a total solar eclipse one can safely image the Sun's corona without a solar filter. The light from the Sun's corona is much dimmer that the surface of the Sun itself.
Unfiltered. ISO 400, 1/500, f/5.6 (EV100 = 12, about the same exposure settings one would use for a subject in full shadow on a sunny day.)

Below is an unfiltered image of the "Diamond Ring" that is visible just as the very first speck of the Sun's surface becomes visible after totality ends. Notice that the exposure is bright enough to see a bright star, Regulus in the constellation Leo, to the upper left of the Sun.

Unfiltered. ISO 1600, 1/125, f/4 (EV100 = 7). FF camera at 105mm (cropped)

On the other hand, when enough of the Sun's limb is visible to see the "Bailey's beads", the sun is much too bright to be imaged without a filter. Below is a filtered shot taken just 24 seconds after the "Diamond Ring" image above! Most annular eclipses will be much brighter than this for the entire event.

Filtered. ISO 400, 1/500, f/5.6 (EV100 = 12 with a 15 stop filter! That's EV100 = 27.) The exposure settings plus solar filter were 20 stops dimmer than the "diamond ring" shot.

So what I am wondering is, if I were to take a landscape photo with a mild telephoto lens (e.g. a 85mm), with an eclipsing sun in a corner, do I still need the recommended protection? Or would a 10 stop, ND 1000 filter be sufficient?

A ten-stop ND filter would probably make things worse. Why? Because many ND filters only attenuate for visible light wavelengths and do not attenuate infrared wavelengths, which is where most of the damage the sun can do will come from. With a ten-stop ND filter, you'll expose for ten time longer, thus subjecting your camera to ten times as much of the Sun's infrared energy!

Personally, I wouldn't include the sun when it is more than a few degrees above the horizon in any frame when using a 50mm lens, much less an 85mm.

It's usually fairly safe to photograph a scene containing the sun using a wide angle lens, say 35mm or wider on FF body, when using a DSLR or other camera that only exposes the sensor or film during the actual exposure. For mirrorless cameras, though, the sensor is almost always exposed just as it would be for a very long exposure and more care must be taken with regard to the sun.

As the focal length increases, more of the sun's energy will be collected by the lens. This means more potential for damage.

Not only can you damage your camera and/or lens, but you can also permanently damage your vision if using an optical viewfinder!

The links embedded above point to other questions we've had here at Photography SE that address most of your concerns. Let us know if you still have questions after reading them.