What helps keep the fee on the Lightning network low?

The lightning network is a great technology and I've been deeply interested in how it works upto the extent that I've previously setup lnd and lightning-app to try out testnet payments on bitrefill and other similar lightning testnet payment gateways. I've had a question on the top of my head which I can't seem to figure out. I believe we all need to open and connect either directly to individual lightning channels or to a channel that's connected to many other nodes. This large channel that we connect to needs to have sufficient funds such that if I'm to pay a huge sum to a 3rd party through that channel very often then the channel should be well funded. Consider there's risk of that channel to keep huge sums of funds and many connecting to that channel to perform transactions, wouldn't it eventually make node operators charge a higher fee as more and more transactions happen at peak intervals? Also, if I would want to run my own node that is connected to many other nodes would I require huge amounts of funds to be open in that channel to earn fees or would it be good to start small and then risk more later?