While I store some coins in them, I still have the majority in my desktop wallet which is Electrum.
What do you mean with '
store the majority in my desktop wallet'? It is always recommended to not have anything on your PC, regardless of what desktop client you are using. Your private keys/seeds should be stored offline, preferably in form of a paper wallet. I'm personally not keen on hardware wallets, but it might provide a decent form of security as well.
I don't think that these online wallets would provide you Segwit-address that could lessen fees as you send funds to other addresses which is a plus for you somehow.
Coinbase is using Segwit, which is cool, but most of them still use legacy format addresses which is a shame. Blockchain stated to roll out support for Segwit earlier this year, but it's clear they are ignoring every user regardless of what their input is. Not surprising at all with Roger and Jihan having a say in how they operate their business.