In this 1944 documentary it is claimed that a netted balloon would work as a parachute when torn, other sources seem to confirm that. Disadvantages are not mentioned, nor can I think of any. Yet it doesn't seem like one can find a netted LTA balloon anywhere today. Also no steerable balloon (or airship) seems to have been built with a net at any point in time. I can only wonder what the deal breaker with nets is?
Balloons have nets! Not only one but several. The nets are just integrated into the envelope.
If you look at this image, you'll see white lines forming a first net.
On a smaller scale, the fabrics used for the envelope are usually ripstop. This type of fabric integrates strong fibers separated by only a few millimeters, making it a second net. The net of fibers effectively stops small tears from spreading across the envelope.
Mostly the consideration will be weight penalty vs risk. Modern envelope materials and construction reduce the risk of a complete envelope failure, such that you would want a separate netting enclosure to perform that parachute rescue function, to a negligible level. This review of balloon accidents seems to show that spontaneous envelope failures are just too rare to be concerned about.
That is, the risk is not sufficient to justify the weight penalty and the handling difficulties of a separate "rescue" netting system (just the bulk of a netting system would require a way bigger trailer to haul around, and its weight would be a lot of grief to handle and move around by a ground crew). It's just too much hassle to bother with for the theoretical protection it provides.
It's similar to parachute rescue systems for light airplanes. Yes they are in use now, but are still really rare in proportion to the fleet, and in any case their existence isn't to cater to structural failures where a plane comes apart spontaneously while flying along minding its own business, but more for flight into IMC situations and such. Parachute systems for ultralights are common because those machines are structurally less reliable and are closer to the "old" balloon model, where you wanted a backup for an envelope that had a significant risk of coming apart by itself.
On the other hand, if you had a rash of spontaneous hot air balloon envelope failure accidents, the risk analysis would change, and you might see a "parachute effect" recovery netting system get developed, using advanced materials, that would be feasible.
But feasible means things like a netting system that is disconnected from the envelope below its widest point, with a gas envelope that can be disconnected at the base, where the heat source is, so that it can move up inside the upper half while "parachuting", that is also light (so a much larger envelope itself isn't required) and easy to handle on the ground.
If you think it through from an engineering perspective, you start to run into technical issue after technical issue, and realize this system could start to get pretty complicated and heavy, leading to the classic "more of this requires more of that, which requires more of this, which requires more of that" engineering weight-complexity-cost merry-go-round. and you're back to "Ah, it's not worth it".