Both parts of what they told you describe how the channels actually work, and the six month minimum is also a retention device. Those aren't in conflict.
SEO means search engine optimization, earning unpaid positions in search results by building pages, getting other sites to link to yours, and establishing that your business is real and local. Search engines are slow to trust new sites, and competitors already sitting in those positions don't move fast. Six months to see meaningful movement in a competitive market is a fair description. It can be longer.
PPC does stop when you stop paying, because you're renting the click. What carries over is everything you learned. The search terms that produced contracts, the ad copy that worked, the pages that converted. That knowledge feeds the SEO work, which is why agencies sell them together.
The reason people do both is cost behavior over time. Paid clicks get more expensive as more buyers enter your market, and your cost per lead drifts up with no ceiling. Organic positions cost the same to hold whether one competitor or ten is bidding. The tradeoff is that you don't control it and an algorithm change can undo months.
Before committing $15,000, ask for the specific deliverables per month written down, and ask which local markets they currently rank in and for which terms. You can check any claim in a private browser window. Also ask who owns the website. If they build it on their own platform and you leave, the pages you paid to rank can leave with them.
