And, what happens, when their nodes are offline? Well, these transactions are never re-broadcasted by anyone. So, when new miners come in, then they don't receive those transactions. They don't include them in their block templates, and they are waiting longer and longer, until some old miner include it in some block.
A new miner connects to a node or a pool that was already online. If a new node comes online they will request mempool from other peers and they will receive the transactions in their mempool which will include the tx pushed by someone who went offline.
Example:
https://mempool.space/testnet/tx/9249f922fbf243ccd1107f9a25db1ed1372dc876b585bdd8f98aab8ad5eea0e5
Says it was first seen 3 days ago but it cannot be found on other explorers
https://live.blockcypher.com/btc-testnet/tx/9249f922fbf243ccd1107f9a25db1ed1372dc876b585bdd8f98aab8ad5eea0e5
https://blockstream.info/testnet/search?q=9249f922fbf243ccd1107f9a25db1ed1372dc876b585bdd8f98aab8ad5eea0e5
Maybe these transactions were pushed directly through their service here https://mempool.space/testnet/tx/push but their node doesn't relay them to the rest of the network. Therefore those txs stay only in their node and show up only in their platform. Meaning no miner sees them to mine them.
This explains why they were mined when you manually download the raw tx from their website and push it using your own node.
Edit:
Here is an experiment to show you don't need a full node and the problem is with mempool.space.
Take any tx from their UI like https://mempool.space/testnet/tx/0ad727680449c102b36baacc1f33cff09da69c42655710af25588632291cf0c6#vout=0
and download the raw hex. Then go to any explorer like https://live.blockcypher.com/btc/pushtx/ or https://blockstream.info/testnet/tx/push to push the tx. It gets confirmed right away like the one I experimented with and was confirmed already after 2 months.