Chainless anchor rodes are possible. Such would depend on: 1. the 'penetration power' of the anchor, and 2. the fluke angle of the anchor. Inotherwords, the 'trigonometry' (especially during the 'setting phase') has to probably be equal to a partly chained rode.
The obvious functional 'advantage' of part chain or all chain rode is during the 'setting' (and re-setting under changing wind direction and changing current direction) of the anchor; the chain holds the anchor stock essentially level with the bottom and thus enables an approximate 15° 'extra' angle (with 7:1 scope) during the 'penetration into the bottom phase' of anchoring.
In all probability one would have to initially 'set' such a chainless anchor with a ~10:1, or more, initial scope then haul in to ~7:1 ... all to get that initial setting angle of the flukes 'into' the bottom as is done with a part chain rode.
The Fortress double angle (choice of 32° fluke angle for 'sand' and 45° fluke angle for soft mud) would probably be my choice for a chainless rode ... just set it at the 45° angle and forget it; but, use an initial scope of 10:1 for sand and 7:1 for soft mud.
(However, Ive found that double angled Fortress anchors sometimes don't always easily 'reset' during ~180° current and wind shifts, as does the heavily-weighted tip 'plow-type' anchors)
Trigonometric (implied free body force diagram) analysis as attachment. The implication is that added chain gives weight/catenary and thus holds the anchor flukes in a much deeper penetration angle, especially during the 'setting' process. The analysis seems to imply a LOSS of ~10° scope angle due to NO catenary during the initial setting/digging-in phase; hence, the probable need to set such with a much longer scope during the 'setting' phase.
Correction to attachment: (TAN ^-1) 1/7 = .17 = 8°
Unless you cover the webbing, from your hawse hole to the connection to the anchor, you can expect very short service life and significant loss of strength due to the long term UV exposure.
When using nylon 'rope', either 3-strand or double braid, the very same section will be noted have quite severe UV (burn) degradation ... but only on the external diameter.
Polyester probably should be the choice because it withstands UV exposure and with less degradation, much better than Nylon(6); polyester tubular webbing is very 'rare'.
The same UV effect is easily noted in jacklines (used with teathers) when long term / long distance sailing when left on deck for a couple of months exposure.
The other problem with webbing is cleating (tends to jam) and handling (hard to grab, and smaller boats this is a manual operation.
At one point I tried using 1/4-inch Amsteel as a leader. I was doing some testing regarding depth of set and holding vs. rode diameter (a smaller rode lets the anchor go deeper). There were a number of problems with this, but the most immediate was braking the anchor out. The stuff is horrible to handle under load, very difficult to cleat successfully, and you had to be VERY careful with you fingers around a load line.
The point is that for a small boat the line has to be comfortable and secure in the hands. Saving weight is great, but the 100 feet of line only weighs 5-10 pounds; saving a few more pounds at the expense of safety (both cutting and fingers) proved to be a poor trade off in my case.
Yes, I could have used steel cable, which could be used safely on a reel, but this was testing and I didn't have a big reel winch handy. In truth, the short Amsteel leader was OK except for breakout. Also, as I mentioned earlier, bare Amsteel was NOT impressive from a chafe standpoint. As an anchoring leader, it is not much better than nylon in this regard. Strange, but true. It's the loose weave. Weave seems to have more effect of chafe than the fiber type.