Once you use a basket a bit it isn't really that hard to get the line in it. I strip with my left hand and keep the basket low and cocked towards my left hip.
I've seen the baskets you posted for sale, but I've never seen anyone actually using on on the beach. It seems it would be harder to keep the line on that easpecially if the wind was blowing versus a basket.
I did a lot of research before buying my first basket and I did wind up with a Linekurv. It's served me well although I might pick up a collapsible mesh or foam basket for the next time I fly somewhere to fish.
I've done a lot of fishing with various baskets over the years. Almost all have their pluses and minuses, as one might expect. I still have the fabled LL Bean basket that the linekurves is based after (more on that later) and have used various baskets from the William Joseph mesh basket mentioned above, to the square Orvis basket to baskets both solid and ones with drain holes...
Mesh basket. Plus for travel, obviously (think suitcase or carry-on when flying.) OK for wading up to your knees, may make a suitable flats basket or out of a boat where a fixed basket is impractical. Terrible for deep wading, if you wade over your waist the line will simply float out (with a floating line) or spill out with an intermediate. Also, the basket in which your running line seemingly tangles the most. My least favorite of the aforementioned styles, but ok if travel and no deep wading are your only parameters, also the least expensive.
Solid style baskets like the Orvis and linekurves... Great for most conditions, including wading deeper in calm conditions. Both have similar cones for aiding in line tangles, both work similarly well. As mentioned, the linekurves is a near identical copy of the old LL Bean model. The linekurves actually is a bit lighter, which I prefer (the LL Bean one is a tank, well made and they were cheap! But kind of heavy...) I like both of these baskets, and for fishing here in Puget Sound, much prefer the solid bottom (no drain holes...) The linekurves is a little larger and a little less expensive than the Orvis, I don't really have a preference between the 2. This style is likely the best "all-around" style, though if your fishing heavy surf (think stripers off of Montauk) then...
Baskets like the 2 above, can be drilled for draining. Be certain to keep the holes small enough that your line can't slip thru. There are also several types of commercially made baskets that incorporate a grate-like bottom for drainage, though the ones I've tried had too large of openings and line could (and would) fall thru, creating a real mess. This is a good type though if you're fishing heavy surf, where crashing waves and spray constantly fill your baskets, often causing your line to "flood" out, as well as being a safety hazard (someone alluded to this above I think...) Great for conditions of heavy surf, ok as an all around basket but again, for our local waters, it strikes me that the solid bottom is a better bet.
Something not mentioned, is simply getting used to using a basket, there are several little tricks that aid in doing so, like making certain your first strip is actually stripped out of the basket (and into the water.) This prevents the line from your reel to not jump up from the bottom of your basket when you cast. Since the line is in the opposite order, if your first strip goes directly into your basket and there is no slack between the bottom stack and your reel, when you go to cast, oftentimes the motion of your reel moving forward and back causes your bottom loops of line to jump up and tangle.
Anyway, that's my thoughts on the subject, good stuff here. For full disclosure, we (Puget Sound Fly Co) sell the linekurves, just got a bunch if new ones in, and can also get any of the other styles mentioned. But again, I'd be hard pressed not to go with the linekurves or Orvis for most situations.
I have been using Rubbermaid Dish Pan stripping baskets for almost 30 years. Cheap, inexpensive, effective. I drill a 3/8" hole in the lip, one at each corner, and I use a 30" bungee cord with a plastic coated steel hook at each end. I do not put anything like drain holes or line tending loops etc., in the basket. I use it clean and bare. It is very simple to flip the water out of it if need be, easy to release if you stumble or trip in deeper water, or take a big wave over the top. And it is a clean dry place for my dirty boots and waders on the trip home. Should you fall into strong current or wash wearing this simple setup, the elasticity of the bungee will allow you to gain your footing, and they are easy to drop off quickly. A solid belted deep plastic stripping basket, even with a mesh bottom, will take on a lot of water quickly in some situations, heavy water, and if you fall in to fast moving water, or you take a wave wearing one of those tubs, you better be good at unbuckling it fast. I have been finding these Dish Pans at yard sales for a dollar or less. And the bungee cords are easy to come by. I still have my first one, which is almost 30 years of use, and I think I paid $2 for it at Kmart on sale, and I found the bungee in the road on the way to the beach. Fishing side by side with other anglers, they using all manner of their own stripping baskets, I get no more nor no fewer tangles or snags than they do. I am astonished at what people are willing to pay for these things. http://olympicpeninsulaflyfishing.blogspot.com
I made three or four of my own and they worked... for a while. The cones I glued on would break or come off and the weed eater line or zip ties would get bent over or warped. I broke down and bought the Orvis basket and it works great, well worth the money.
For the boat option I use a tall laundry basket and it works great.
I cut the William Joseph netting off the belt and put the Ikea footstool where the netting would be. This combo works really well. The belt is very comfortable and the basket works well. Good idea for the berry picking!
I have been using a KR Line Tender(collapsible stripping basket) by HMH for many years while fishing Baja beaches for sierra's and rooster's. For the past four weeks I used it almost every morning in 1 to 3 foot waves with no issues. I have modified the basket using 1/8 inch painted plywood to form fit the bottom. Corn cob handles are attached to the plywood which usually eliminate line tangles. The plywood forms a solid bottom and helps the sides of the basket to be relatively rigid. The collapsible basket fits into a 10" x 15" x 1" mesh bag. It takes up very little space when traveling. The collapsible basket was pretty useless without the plywood and corn cob handles.
I mostly fish out of a boat on Puget Sound but often go ashore to fish gravel bars and shallow shelves plus occasionally walk 1/4 to 1 1/2 miles into a couple of beaches. It is very convenient to carry the basket slung over a shoulder when walking. In my boat I use a collapsible cloths hamper with two 3/4" treated plywood circles to keep the basket from blowing out of the boat when motoring. Corn cobs handles are again attached to plywood.