. I really just need a piece of software like we used to buy in cardboard boxes off a shelf, for a single user on a single computer.
Given your requirements and unusual preferences, I'd suggest you install MSDOS on a PC, and find a property-management package written in the 80s.
Seriously, there are businesses that do this, and I can't fault them for it. When you have something that works, leave it alone. There are MANY businesses still with that old MSDOS computer sitting in the corner chugging away, printing some kind of reports on a dot-matrix printer.
On the other hand, some law changes, or just the nature of the business changes, and now you could have a BIG problem. The software probably isn't published any more. If it was custom, the developer has almost certainly moved-on and won't be interested in supporting it, or else is in the Old Age Home for Programmers.
I can't imagine why a landlord would prefer paper checks to ACH. Renters often either forget, or else "forget" to send paper checks. Many will put ACH on auto-pay.
I think that in general you are wrong about the motivation for monthly service fees for software. No software is ever "done". The ongoing maintenance has to be paid-for somehow. If not a monthly fee, then a yearly one, whether "in the cloud" or not. The old model of paying for software created significant problems for publishers. If you do only a "support fee", then customers stop paying until they have a problem. And then they create reputation problems when they try to use their 5-year-old buggy software (or that simply doesn't work with the newest browser or OS - which is something the publisher could not have possibly accounted-for) and then complain, and the software gets a reputation for unreliability.
MOST problems with locally-installed software are due to: amateur set-up of workstations/servers. Not keeping OSs up to date. Workstations/servers (especially Windows) getting infected by viruses. Inadequate operational knowledge. Inadequate backups. No off-site backup.
"Cloud" services eliminate all or nearly all of these concerns.
On backups: on-site backup is simply a non-starter. If you are going to do local backup, you need to take a backup at whatever interval makes sense for your application, take a copy to a safe-deposit vault, and ship a copy to another part of the country. In fact, I actually did this in the early 2000's for some high-frequency-trading software. We fed-ex'd drives between NY, St. Louis, and San Diego on a rotating schedule. Today, cloud backup with local backup as a convenience makes much more sense. The local backup would be the first choice if you need to restore, as it would take significant time to retrieve a remote backup online, or a fee to have a hard drive shipped.
For cloud backup, you need to insure that the data is locally-encrypted before sending off to the cloud. If good encryption is used, and somebody decrypts it, you have bigger problems - because then it's the NSA that decrypted it!
You will find very few vertical software publishers any more that are interested in selling installed software.
I will ask my friend, though, what property-management software he used before his employee unceremoniously destroyed the hard drives!
And, FWIW, surprise, surprise (cynicism) an add for apartment management software just appeared on this page. I'll include the link only for information purposes. I know nothing about it:
I was unable to determine from their web site, though, whether this is installed software or a cloud service.