Running Core as a wallet

Hey all.

I was recently in the market for a new cold storage wallet, and then I kept seeing tweets about how if you aren't using your full node to validate your own transactions, it is essentially useless (I know it's not entirely true). It got me doing some research about using Bitcoin Core to store all my Bitcoin as it seems like this really contributes to Bitcoin's decentralization.

I also know that I could use a lightweight client (I assume this would be the cold storage computer UI) and connect it to my full node and it offers a similar advantage.

Please help me understand what I should be considering here.

Additionally, would it be safe to store all my Bitcoin on my core node?

Thanks in advance.