About SEO Outreach...Where to send it from?

I assume that agencies or freelancers who provide SEO outreach services will have to use their own tools or capture potential link prospects in an excel sheet and then send a bunch of emails asking for backlinks.

Just curious, do you think it might be better if the email were send via the client's domain instead of the agency's one?

How do you guys do it?

For example, if someone were to reach out to me with a Brad@abcagency.com to ask me to link to a website DEFurnitures.com, I probably wouldn't bother opening it or replying, but if it were sent via brad@DEFurnitures.com, do you think the chances of getting a link or at least a reply back would be higher?