I did the same thing on two homes that I own. The first one was a one-story ranch style home. There were some problems and damage to the existing gutters and downspouts, so before replacing them I decided to just remove them all and see what happens.
A neighbor said that if I did that I would end up with deep ruts in the ground along the line where the rain water drops off the roof. He was somewhat right, but not completely. Along the back of that house, there are some narrow ruts or pits in the ground but nothing significant. Along the front of the house, not so much, but the grass grows better in the front than the back which reduced the amount of "rut effect". And, over the back door there is the problem of having to walk through a waterfall while going in and out if it is raining. I may be able to create some type of fix for that issue by putting up some kind of diverter piece on the roof to divert the water away from the section over the door, but I haven't bothered to try that yet.
The second home that I did this on is a 2 1/2 story older style home. That one has a huge oak tree in front that belongs to the County so I cannot remove the tree. The tree produces tons of leaves and acorns and always created a huge mess with the gutters and downspouts. Interestingly, for that house, the issue of the "ruts" is almost nonexistent. I think that may be because even though some of the waterfall drops from two floors up, the amount off roof area receiving rain on each of the 4 sides is less so less rain water pours down on each side.
Maybe something like that would be a "fix" of some sort for the rut issue that my no-gutter, no-downspout, solution created.
But, the bottom line for me is that for these two homes -- which are subjected to tree leaves and acorns etc. -- my no-gutter, no-downspout, solution seems to work well.
And, for homes where I have no tree issues near those homes, leaving the gutters and downspouts in place is no problem and works well.