In my experience, it’s rare for one website company to take on the work of another.
Developers hate working on what we call “legacy systems” – systems with a history we aren’t fully aware of and that we can’t be 100% sure is properly documented.
Taking on a website from another developer/agency and providing support on it is like taking on the warranty for a second-hand car that you’ve never worked on before with no service history and the strong suspicion that that rattling noise under the bonnet is probably something serious.
The reality is that most migrations from one website agency/developer to another involve a rebuild of the website.
This doesn’t mean that it is impossible and this is a strength of working with one of the larger Open Source platforms such as WordPress. There are lots of WordPress developers around getting one to take on your website shouldn’t be that hard – just don’t be surprised when the suggestion of a rebuild comes along.
Be honest – do these guys just want my money?
In short – yes, of course they do.
The longer answer is that what they want, like any business, is profitable contracts.
Taking on a site built by someone else, even if you understand the technology in detail, always carries a risk. The value of the support contract, therefore, has to mitigate that risk, or the person taking on the website is potentially going to be losing money. Replacing the website is therefore more cost-effective; a higher project cost but lower ongoing running costs.