first off welcome to the site
MY first question for you is, in your picture you provided, your bucket is NOT on the ground, it that how its sits when you have boom all the way down
If so, then I think you do have a wrong bucket for your machine! or there is something wrong with your machine!
NOT all buckets fit correctly per every machine,(most are close though) the height of where where the quick attach is, can vary from Bucket to bucket, which then, down or does allow bucket to sit flush on bottom with the ground
you can measure things, by placing both machine and bucket (off skid steer) on a level smooth section of say concrete pad, or pavement or even GOOD level dirt
and then, measure from ground to top of your skid steers quick attach, and then, measure top of attachment on bucket
, and see what they measure
and buckets come in many shapes and sizes, and service ratings(l like light duty/ standard, heavy duty , and severe duty are the most common) from MY experience, many tracked machine, use what is called LONG bottom buckets , so the operator can see the edges/ends of bucket better, which also helps them dig in better from my experience!(and not claiming to be any expert here)
your machine should have something like what is called a FLOAT option, which allows the bucket to sort of lay flat on the ground with neutral pressure skim the surface!
your owners manual should show you how to do this,
NOW< if your land was once all FARM or AG land, that top layer of dirt(AKA top soil) is typically a valued product, and can make you money
if your going to remove a lot of it???
it might be worth talking to anyone in your area that sells top soil
as they are typically always looking for new sources to get it, and they will come and remove it for you and PAY you to do sit
saving you wear and tear on your machine and putting money in your pocket over you spending it ,
NOW< if your wanting to SAVE this top soil for future projects, well, then
thats a different game!
and honestly, if your trying to stop mud, I would maybe just suggest seeding it with something that hold the soil in place, having a lawn full of rocks, is never fun in the long run, so removing the top soil, might bit you in the long run, all the more so if you live where it freezes and thaws, which will push rucks up all the time, once you remove the top soil)food for thought here)
I don't know how much of an area you want to remove the top layer from, so, its hard to says whats best for the job without knowing this
and not knowing how HARD the top layer is,
but typically one can lower boom, slightly angle bucket down, to get it to DIG into the ground, drive forward some to get bucket to STAY under things, that is if ground is not rock hard, if its very hard some times you can angle bucket more and lower boom a little , as if like you wanted to LIFT the front of machine using bucket, so the edge of bucket get DOWN force using weight of machine to allow it to DIG into the ground
then once you get bucket down to where you want, you want to start to level bucket out and drive forward , as you go forward if it wants to climb up, you tilt bucket back down or lower boom more, what ever is needed to keep edge of bucket in the ground at your desired depth , or what you can get,
some times you have to take things slow and only a little level at a time, making many passes to get to desired level!
harder the ground the slower going things will be, , your skid steer is NOT a bulldozer, so at some point your going to likely run out of traction, pending again how hard and heavy your soil might be,
NOW< if you wanted to maybe RENT some other implement, as opposed to buying something, so you can see how it works and or you like it or not, which is why there are rental shops!
a soil conditioner is a good thing to look at IMO for this type of work
it will chew up the top layer making it loose for you to scoop up easy
and they can be easier for some to get down to "X" inches better than just eye balling things with a bucket, plus some have ability to angle as well, which your standard bucket will not(hi tech ones will have laser levels you can use for exact grading too)
or maybe,
as mentioned, a Dozer blades are out there as well, but I think they are not as easy to control depth of things as a soil conditioner is, plus a soil conditioner will leave the ground ready for seeding, and get things growing faster, to help resolve wet mud contortions for you!
a toothed bucket will dig better than a smooth edge one, or even a bolt on set of teeth, but I think your better off r renting a better tool like above suggested ones!
MOST good rental shops have these things, if your willing to look around and try one!
you can surely also, look at youtube and see them in action and go from there too, post back what you go with if you do, always nice to hear results from posts!