You colony is too small to live normaly and get honey. Swarms has been about 1 kg. If swarm is 4 kg, it works normally honey.
You are right that part of old bees have died. When you take a little swarm, usually queen is not mated and it take 10 days before it begans to lay eggs. Then it takes 3 weeks before first bees emerge from combs.
During that 4-weeks half of bees have died for working. So it takes 5-6 weeks before colony grow bigger.
Next stage is that colony occupy a whole box. Queen fills it with eggs. If there is a lot of honey, it makes tight in one box. You can give another box under the brood. Bees enlarge their living downwards if they are able. So hey do in nature. Do not give empty box over the broodbox. They gather honey and colony lives better when they get new workers.
If you want honey, you should get 1-2 kg swarm and combine them.
When you have swarm queen, such a colony inherits terder to swarm, and it swarms next summer. It is better to get commercial queen, which is so eager to swarm.
Now your colony is at same stage as many hives here in Finland. It takes 1,5 months and your hive is able to get honey normaly. Then it will be 3-4 boxes or more.
You you could have even 1 kg swarm, it would be able enlarge and gather honey at once. 2 kg more to recent would be very good. It depens also how vegetation developes.