If you have to thoughtfully, artificially compose emails, they won't be authentic anymore but fake, radiating faux "warmth"
There's not much value in authenticity if you're cold, lack social awareness, and you can't compel others.
The first step to learning to be warm is to practice being warm so that it's eventually authentic.
That said, TFA's examples are particularly saccharine. They remind me of when you're talking in good company and someone keeps feeling the need to use preambles like "I totally know where you're coming from and what you said is completely valid, I don't want to step on your lived experience, but let me come at it from a different angle with [...]" and I'm thinking "my god, just fucking say it."