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.
Chain does help with the initial setting process. As you point out, the catanery is beneficial. This is particularly important in difficult substrates where the anchor may be struggling to achieve the initial bite.
However, in the later part of the setting process the catanary is almost lost, as the force essentially straightens out the chain.
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.
With a good modern anchor and a reasonable substrate, there is no need to go with very large scopes to start the initial set, even with zero catenary, but with poor substrates it is a tactic worth considering.
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)
If the fluke angle is too shallow for the substrate (so using 32° when the substrate is soft enough for 45°) the anchor will still work fine, but the holding power will be less than at the 45° setting. On the other hand, using a fluke angle that too steep (so using 45° when the substrate is too firm for 45°) the anchor will never set, or if it sets it will rapidly break out and the holding will be very poor. You therefore need to be very careful using these wider fluke angles. You need to be sure that the substrate is soft (and it needs to very soft to use 45°) where you will be dropping the anchor.