I never use a climbing screw to launch on frozen lake. if you can relaunch the kite you can launch it on ice or the soccer fields just dont screw around so long that the kite slides away. I know plenty of people who love the fixed attachments, if you like them, I'm sure you can build one for the snow, maybe bring a big bag of rock salt, tie a rope to it and clip on to that. Later in the day, you can use the salt on your sidewalk . . . .
its the same as if you were self launching on the beach with out a fixed anchor.... even easier on snow actually...
If you dont know how to do it you really should learn... plenty of videos out there. One really cant consider themselves a self sufficient kiter if they cant launch and land by themselves with no outside help (including a fixed anchor)
Dont get me wrong- i like the anchor method alot myself... but if something happens and you cant get back to that point you need to know how to land your kite... if something happens mid downwinder and you need to re rig and re launch you kite you need to know how to do it completely by yourself....
I can self launch and land with out an achor with a patch of sand not much larger then my kite- I dont like doing it... but it has saved me from a long walk on a couple occasions...
Just be careful as you probably already know- kites slide alot easier on snow and ice so they need more weight to keep them from blowing away
+1 on NO ice screw! No snow stake!! No metal!! I teach kiting as well as ice climbing and really many ice screws are not designed for the up and out pull the kite can put onto the anchor point. So you can very easily break the screws hanger and or bend it.... Same with the snow stake only worse as there are no relief threads.
Another skill to perhaps focus on is emergency take down of your kite. On any surface with any kite.....which I must say has nothing to do with ice screws...
Look into the " v thread" or "a thread"
As a guide, I use this more then ice screws and they are not only stronger, but they don't leave metal anywhere. But again on snow, they don't work.
In the winter I always use a Icescrew on the ice and a iron stake on fields. Nice to just hook the kite to the anchor and have a beer. When riding with others hook the kite itself to the anchor instead of weighing it down. Seen many a kite go Bye Bye across the lake, miles away to end up ripped and shredded when it hits a bush, tree, Shanty or shorestation.
Another alternative might be to install a 5th line on your kite and launch/land that way. I've been using my 5th line to launch/land solo on snow for a couple years... use my snowboard as the anchor... flip it upside down so the bindings keep the board from sliding... there's hardly any pull from the kite when deployed on the 5th and that's the reason it works.
It might be a tall order to install a 5th line on a 4 line SLE but I've heard of people doing it... you'd remove the line when you hit the water in the summer of course.
On bare ice with no snow, good luck trying to self-launch without some sort of anchor point. You've got nothing to weight down your kite (unless you carry something with you out onto the ice), and even a little wind will cause your kite to drift downwind like crazy. I've often needed an anchor point just to pump up my kite without sliding downwind. I use an ice screw where I launch, and carry another one that's easily accessible in case I want to take a break and I'm not near my launch spot. I use a switchblade on my snowboard, so I can ride on bare ice very easily.
Self-landing, in an emergency, is no more difficult than on water or the beach. Either flag the kite or flip it. My newer kites flip over when you pull in the front lines. Older kites flagged on one line. I just pull myself to the kite using the line(s).
Getting back on topic, self-launching on snow on land can be difficult if you can't find enough loose snow to weight down your kite, and there are no convenient anchor points (fence posts, sign posts, trailer hitch on the car, etc). In those situations I often carry a small bag of sand (20 pounds?), and then self-launch using the walk around/warm launch technique (lines laid out downwind of the kite, slight tension on the lines, walk around in a big arc until the weight flops off the kite and the kite launches.)