Apple people, and people who have shifted away from "texting" to using some other platform (e.g. whatsapp) often shift between computer and phone for this job.

But in the US at least, people outside those groups typically don't. I don't think you can (easily) send SMS from a laptop as though it were your phone.

(Yes, it could be argued that the two groups in the first paragraph are actually ONE group, since Apple "texting" is silently-to-the-user shifted to iMessaging, i.e. another app.)