What if however I wanted to send a transaction from my own node and the mempool is already clogged up beyond the 300 MB cap for most nodes with other transactions (maybe with inscriptions, ordinals etc.)?
Either you wait for the mempool to empty, or as DaveF says, you increase the fee and your transaction will evict some other lower paying transaction from these nodes' mempools.
I think nowadays the default amount of nodes that your node connects to is only 10, right?
10 outbound connections by default, but up to 115 inbound connections by default. Note that apart from the 2 block-relay connections, outbound and inbound connections are functionally identical - it's only the way the connections are established which are different.
What if this is some sly roundabout way to allow transactions to be made on CEXs/authorized transmitters or something like that?
It isn't. Anyone can increase their transaction's fees and gain access to a mempool which is already at its limit by evicting other transactions. Since mempools are run locally, it is always in miners' best interests to do this, since it allows them to maximize their profits.
I did transmit a transaction from my node (version 20.2) and I can see that there's been 6 blocks added to the blockchain since my transmission was sent.
Looking at the time this happened, you simply got unlucky with when blocks are mined. Keep in mind blocks are mined
on average every 10 minutes, but this can vary from a few seconds to over an hour in reality.
Block 812,957 included transactions with a fee of 3 sats/vbyte. At this point, mempool.space was probably recommending a fee at around 5 sats/vbyte, and appropriately so. However, it then took 42 minutes to find the next block, by which point the minimum fee had increased to 24 sats/vbyte. Because of this backlog which was created during this 42 minute time period, it then took the network to block 812,970 to mine transactions at 5 sats/vbyte and include your transaction in a block. This had nothing to do with mempools being full - indeed, if nodes were rejecting your transaction then it would never have been mined at all. This is simply a quirk of how bitcoin works and it happens all the time.