It is probably the greed in the big mining companies to take advantage of the current pump by clogging the mempool by not allowing any low sat fee transactions to go through into any block and taking only high sat fee transactions for confirmations.This gives the air to the greed of other small miners too, and they also set the minimum fee higher to attract high fee transactions in their blocks.
I don't think you understand how mining works. No miner is clogging the mempool. No miner is setting a higher minimum fee. All miners are simply maximizing profits (as any sensible person would do) by taking the highest fee transactions first in to the limited space in each block.
correct me if I am wrong but does this mean that the reason why the transaction fee increases (other than people flooding the blockchain with transactions) is that there are not enough miners to solve blocks every ten minutes to keep the transaction traffic to minimal?
No, not at all. The mining difficulty is readjusted every 2 weeks to keep the block time to an average of 10 minutes, despite fluctuations in hashrate. If suddenly half the mining power disappeared from the network, then things would be slower for at most a few of weeks, but then the network would readjust to the new hashrate. At the moment we are pretty stable actually, and over the past 2 weeks are only running around 4 blocks (40 minutes) behind where we "should" be, on average.
Participating on Level 2 requires a transaction on Level 1 first. The system can't accommodate everyone's participation, nor will many people be able to afford participating.
The system can accommodate it, and people can afford it. To open a Lightning channel on the base layer requires a single transaction. Once your channel is open, you can theoretically make unlimited transactions through that channel, being limited only by the amount of funds you have in the channel. I can open a channel with 1 BTC in it, paying a few cents in fee at a rate of 1 sat/vbyte, and then I can buy myself a $5 coffee every day for 15 years, making 5,600 transactions on Lightning, all without making another transaction on the base layer.