How can I add a domain just by changing the A record and not the Name Servers?
Well, you literally just point the A (address) record to the web server that'll be hosting the website. That's what web browsers use to translate a domain name to the server's IP address, and most of the time that's all you need. (Though if your hosting company cannot guarantee a fixed IP address for the web server, you'll want a CNAME (alias) record instead, which is slightly more troublesome.)
However, switching the whole domain's nameservers to use the web hosting company is completely overkill most of the time, and can be very disruptive to your client's existing IT systems. Recall that the 'nameservers' setting specifies where DNS records are hosted, not where the website is hosted – it is a completely separate service from web hosting.
Companies might already have other services on their domain (e.g. email, or internal servers, or other subdomains in general) – i.e. they already have a lot of custom DNS records under their domain, and switching nameservers would mean they'd need to move all those records to your hosting company's DNS panel as well. That means the client's IT team would need to go through you in order to make any DNS changes, even those completely unrelated to the website.
In addition, many web-oriented hosting services have very basic DNS hosting and just outright do not support features like less-common DNS record types, DNSSEC, automation via APIs, split views, etc. Unless you're 200% sure that the new nameservers can provide the same functionality that the company uses on their old nameservers, you cannot insist on this change.
I've worked with over 200 companies before and this is the first time I've had someone tell me this.
Those previous companies either had no existing use for their domain whatsoever (so no existing DNS records to lose), or they obtained a fresh domain for the website (i.e. it wasn't hosted on their "primary" domain).