(Flash from the "where in blazes did this guy come from?" department...)
If you have a DH Private Server and you are using your required shared account for Apache, PHP, MySQL, etc, then you are still subject to all of the issues of shared hosting.
If you are in a DH PS and you want to get full isolation from us shared riff-raff folk - that is, you don't want to use the DH Apache, PHP, MySQL, etc - then you need to maintain all of this yourself, no? Seems like a lot of admin responsibilities for someone focused on apps.
Like the OP, I'm not a qualified LAMP admin, nor do I have the time for that sort of work. Since a site is required to have a shared space in addition to a private server, I personally would host multiple client domains in the shared space and only call back to private space for special things that one can't get from shared space. For example, I would get a DHPS for alternative databases, Mono, Java, testing OSS Enterprise apps, general Linux development, and low-profile persistent processes/services that might serve as a back-end for shared web space. If I have more clients that will fit comfortably in a single shared space, or any client prefers to have their own secure space, then I would get them a separate shared account but still make use of my PS environment as an app server.
The only exception to that is a site hosting apps (game forums, do-it-yourself blogs, Ecommerce) where better performance is desirable and the client is willing to pay for it. In this case I would consider one or more MySQL PS environments just for that. As long as DH is managing this, there's no headache and supposedly the performance should be much better.
Have I stated anything here that doesn't fit the reality of this offering? I'm seriously considering jumping on soon too.
Regarding porting over clients that are currently paying $40+ for private services elsewhere: If you're managing their environment, then it doesn't matter if you have them in a $6/month shared server or a $15/month private server. They are paying you for your management services, and your services are worth some $/month. If they don't want site management they can manage their own host and see how much it costs them. Whatever you do, make sure that the services that you purchase don't cost you everything you're getting paid in maintenance time. If providing services becomes time-intensive you will be losing money because it's time you're not spending writing code. Also, document a firm policy for exactly what you intend to provide your clients for the money they give you. Some clients pay the fee and you never hear from them. Others will consume 40 hours per month for the $40 (or less) that they're paying you. Establish cutoffs, additional fees for extended efforts, and clearly define your responsibilities, otherwise it will suddenly be your responsibility to fix everything that happens on the internet - in the client's mind and perhaps legally too.
If all of this is accurate, maybe this is wiki material? If not, I'll humbly crawl back into my shared hole.