I think this is a good idea. With few claims to start, the priority is always going to be on ensuring you can upgrade. With more claims to start and fewer to be gained immediately from upgrading, you'll have more viable options and more opportunity for making both good plays and mistakes.

I'd take this even further, though. As it stands, claims are the one sure path to victory, and upgrades are the one sure way to get claims. So even if you make L2 and L3 provide fewer claims and/or make them more expensive, the right play is probably still to rush to L5 as rapidly as possible. What's needed is to make upgrading to L4/L5 a viable strategy, one of many, rather than a near-requirement for victory. There need to be serious trade-offs to upgrading, and there need to be alternate paths to victory.

One that springs to mind is encouraging players to get into the stock market while still at L2/L3, so that buying out rivals becomes a viable way to increase your claims. And at the same time, reduce the number of claims you get at L4 and L5, and jack up the costs of the L4/L5 upgrades.

My concern about giving away too many claims on founding is that that would give players who found first too big of an advantage because they could claim all the high tiles right off the bat. (And, yes, I know that there is a problem with Expansive getting to Level 2 too fast right now...)

I really, really like the idea of one claim drone at a time. It introduces additional important and difficult decisions.

Maybe two at a time so that players don't get too frustrated waiting for something to happen right out of the gate.

Something like this is my big worry, too.  The suggestion of one claim drone might help, but even there you're going to see a gap.  I mean, the drones aren't that slow, and first-to-last founding times can easily be twenty or thirty seconds, enough time for the first drop to spend all of their claims (even with an expanded number of them) before the last player drops.

What if claims worked slightly differently?  What if your HQ determined your maximum number of claims, and then every so often (some sort of fixed schedule), everyone who is eligible gets a new claim given to them?  This would reduce the pressure of needing to upgrade fast, but make claim order very important.  Not sure this would actually work, or even necessarily be fun (I mean, if the interval is 1 minute, what are you supposed to do during the minute between making your first claim and waiting for your second one to become available?), and it might even further emphasize the importance of starting locations (if two players both spend their second claim trying to claim a high water tile, whoever is closer to the tile will get it because they get their claim simultaneously).

Note that this MIGHT also help with the extra claim from founding last bonus.  If that extra claim is just tacked onto your maximum number of claims, then you basically get your extra claim as late as possible as you would get an "extra" allotment after everyone else has gotten all of theirs.

EDIT: Just realized that this would also have a knock-on effect for black market claim-buying.  You'd either need to get that claim immediately, vastly increasing its value, or it would add to your "maximum" which probably decreases its value as you have to wait for it.