Would I get a washed out photo on a wide angle lens without the solar filter?

Without a filter the sun is roughly 14 stops brighter than the highlights of the landscape under it on a sunny day. You must make a choice: get details of the sun (with at least some kind of fairly strong filter and darker exposure value) OR get details of the surrounding landscape while allowing the sun to be completely oversaturated and cause a fairly large area of the sky around it to also be washed out.

As the answer in the other question you reference states,

... NO manufacturer of cameras or lenses has ever said anything to the effect of, "It's okay to look at the sun through our camera's viewfinder." If in doubt, use Live View.

It's also likely the case that no manufacturer has ever said, "It's okay to point your camera at the sun without any kind of solar filter. Some of us do it on occasion, but we do so at our own risk.

I've taken plenty of wider angle photos (say between 17-24mm on a FF camera) with the sun high in the sky and in the field of view without any detrimental effect on my lenses, sensor, or vision. But the sun will not be clearly visible in such a photo if it is exposed for the terrestrial scene. If the photo is exposed so that the sun will not be overexposed then the rest of the scene will be very dark or even black. There's just too much difference in brightness between the sun and the ground to capture both at the same time unless the sun is very low near the horizon.

This frame was shot at 24mm on a full frame camera. ISO 100, f/8, 1/320 second. Had the sun been properly exposed it would have been about the same size as the full moon when you use a smartphone to take a photo of it. Even as it was, I shot the image with the people in the scene very underexposed and then pushed the shadows to near the breaking point in the raw conversion process.

Both of the following images were shot on a FF camera at 17mm. The only difference between the two was the exposure: Both were shot at ISO 100. The first was taken at f/9 and 1/200 second, the second was taken at f/8 and 1/800 second. The difference in the size of the glare from the sun is due to the 1 2/3 stops difference in the overall exposure. Even in the last one the diameter of the sun, at about 4.3% the length of the diagonal, looks roughly 8.6X larger than its actual angular size of 1/2° in the sky - which is roughly 1/200 (or 0.5%) of the 104° diagonal angle of view of that lens at 17mm on that camera.

Note: In the first and third images above you can see a 'ghost reflection' of the sun at approximate actual size directly across the image center from the sun. This is a reflection off of lens elements. Since the sun is high and slightly to the right in both images, the reflections are low and slightly to the left in each image. Anti-reflective coatings in modern lenses are very good, but when you have a light source that bright compared to the rest of the scene, you'll likely see such a reflection.