“When you use the hardware wallet it is connecting to the ledger/trezor servers and telling you what you have in those addresses. You can’t actually verify what they are telling you is the truth. If you run your own node and connect that node to your wallets, you can verify the truth.”
But that is not the only reason. The fact that only economic full nodes can help enforce concensus is perhaps just as important:
“Running a full node is kind of like voting where the weight of your vote is the amount of money you transact through that full node (sort of).”
In other words, when ur full node is not connected to ur actual BTC stash, u are giving up ur leverage to enforce consensus on the bitcoin network. Many don’t realize that running a full node with an empty wallet does nothing for the network. It is completely redundant.
That happens a lot these days because bitcoin core’s compatibility with hardware wallets is minimal. U need to run code to connect it, and even then it is cluncky, there isn’t even a GUI. It also happens because folks don’t know about the importance of economic nodes.
Here it is from the wiki:
As explained previously, full nodes enforce the consensus rules no matter what. However, lightweight nodes do not do this. Lightweight nodes do whatever the majority of mining power says. Therefore, if most of the miners got together to increase their block reward, for example, lightweight nodes would blindly go along with it. If this ever happened, the network would split such that lightweight nodes and full nodes would end up on separate networks, using separate currencies. People using lightweight nodes would be unable to transact with people using full nodes. If all businesses and many users are using full nodes, then this network split is not a critical problem because users of lightweight clients will quickly notice that they can’t send or receive bitcoins to/from most of the people who they usually do business with, and so they’ll stop using Bitcoin until the evil miners are overcome, which is the appropriate response. However, if almost everyone on the network is using lightweight nodes in this situation, then everyone would continue being able to transact with each other, and so Bitcoin could very well end up “hijacked” by evil miners.