A web wallet will be nice to have certainly, but I would say it is a more ambitious project, for the reasons you mentioned, and more. It will likely entail compiling some or all of the neptune crates to wasm, for starters.
I think also that web-wallets for other cryptocurrencies typically do not attempt to run a full node. think of syncing and storing all the blocks in a browser. Instead they would connect to a full node somewhere. So this underscores the need for an architecture that moves wallet out of neptune-core, so that it can concentrate on managing the blockchain, and serving various wallets of all types. But that architecture change is itself a substantial project.
In short, at present a desktop wallet software seems the most achievable to me. Though it would also benefit from a modular architecture, so to my eyes, it looks better to delay the competition until such an architecture begins to take shape and there is at least a proposed set of APIs for such an independent wallet to use.
crawl. walk. run.
horse, then cart.
etc.