Electrum wallet is the best option. You can see this link to learn how to download the wallet and verify the signature
Signature verification is a step to ensure that you downloaded programs from the official link. The developer signs the program code using a digital signature, and if a single line of code is changed, the signature verification step will fail.
Also, wouldn't Bitcoin Core client start syncing the blockchain as soon as I start it?
Yes, but the data will be downloaded to the default directory, which is
~/Library/Application Support/Bitcoin/
You need to modify it to the external hard drive data path.