Are there any pools that accept transactions with a fee which rate is fewer than 1 sat/vbyte, perhaps? Or do they all just reject such transactions even if mempools are empty?
Minimum miners are paying these days is 1 sat/vB, but we usually see in mempool.space is something around 5 sat/vB, so I would not risk going with bear minimum.
Back in the days it would be possible to sent bitcoin transaction without any fees, but mining today is more about profit.
You can however always start your own mining operations even with solo miner, but don't expect it to be very profitable.