Is our lightclient slower than PAPIs? Let's check!

This comment came up on our Matrix Subxt Lounge room:

I played a bit last year with yew and subxt using the light-client feature, end up doing this small dapp https://www.goclaimit.app/?chain=polkadot&lc=true

Compared to PAPI, the app takes much more time to initiate the smoldot client before it can be used to fetch data example https://bounties.usepapi.app/

Let's check this; we both use Smoldot so I can't see a good reason why one would be any different from the other here.

One way to check this is to make a Small Subxt example using the lightclient to do something, and make the same example in PAPI, and then time how long they both take. if there's an obvious difference then we'd need to work out which part of the script is taking the time and why.

Start by making the Small Subxt light-client example described in the issue and an equivalent PAPI example, then time how long each takes to initialize before fetching data. Compare the results and identify which part of the script accounts for any obvious difference; done means the performance difference and likely source are documented.