Matt's answer covers some good ground, especially this line: "Odds are they chose Craft and their custom plugins for a reason and those reasons are still vitally relevant, as is the ability to stay nimble in the future."
Wanted to expand on some points...
I've walked into these kind of building-on-fire situations before and it's important to figure out the history of the relationship before you start. That will inform how you tailor your approach.
- Why didn't the previous firm work out? (Were they nickel-and-diming the client for example? Or did they not frame after-site expectations properly? etc.)
- What does the client actually value? (Or was it a buddy higher up helping a buddy situation?)
- Has the client set any money aside for on-going maintenance?
This scenario is all-too-common and almost a cliche. There's too many fly-by-night agencies/developers that are onto the the next client as soon as the job is done and do not support what they created.
There's less money in support than creation so I understand this aspect of it. But knowing where the client is coming from is helpful. Designers blame the client but 9/10 the client is not at-fault—that is, if you sell properly.
I have to credit Ben Parizek for this quote on the Control-Click Podcast but he's right on the money when he says "we're not just building the client one website, we're building them two websites." In other words, the backend of the site is just as important as the front-end.
And that means on-going updates, maintenance and hosting. Clients (and some designers) are still treating websites like printed pieces where you build it once and it's done. It's more like a living, breathing entity. The web thrives on change. If your business objectives ultimately don't involve changing over time, there's not a whole lot you can say to convince the client otherwise.
Clients also generally don't care what technology you use, they want solutions. If they feel your solution doesn't provide them with any more value than a SquareSpace or Weebly site, than you've not framed your value to the client properly.
What did those plugins actually do in the first place? You don't get custom business logic for free. It's like building a luxury jet; you may have paid for the custom GPS system, leather, stone tile, and jacuzzi, but you still need to fill it up with fuel to fly it, mechanics to work on it, etc.
Lately, I've been building in some sort of maintenance agreement with all of our site builds for a period of time (including support, hosting, and CMS updates, etc.) That helps solidify trust with the client that you're not going to leave them out in the desert and helps makes sure their business objectives have been met. Ultimately after that maintenance period is up, it's an easy renewal contract to keep them in, assuming you've set that initial expectation already. It can often lead to bigger projects, too...