The DC will handle physical service for you if something breaks, you just pay for parts and labor.
All of this requires knowledge, of course, but it’s hardly an impossible task. Go look at what the more serious folk in r/homelab (or r/datacenter) are up to; it’ll surprise you.
> I'll have to monitor more things (like system upgrades and intrusion attempts)
You very much should be monitoring / managing those things on AWS as well. For system upgrades, `unattended-upgrades` can keep security patches (or anything else if you'd like, but I wouldn't recommend that unless you have a canary instance) up to date for you. For kernel upgrades, historically it's reboots, though there have been a smattering of live update tools like kSplice, kGraft, and the latest addition from GEICO of all places, tuxtape [0].
> I'd also have to amortize parts and labor as part of the cost, which is going to push the price up.
Given the prices you laid out for AWS, it's not multi-AZ, but even single-AZ can of course failover with downtime. So I'll say you get 2U, with two individual servers, DBs either doing logical replication w/ failover, or something like DRBD [1] to present the two servers' storage as a single block device (you'd still need a failover mechanism for the DBs). So $400 for two 1U servers, and maybe $150/month at most for colo space. Even with the (IMO unrealistically low) $200/month quote for AWS, at 5 months, you're now saving $50/month. Re: parts and labor, luckily, parts for old servers is incredibly cheap. PC3-12800R 16GiB sticks are $10-12. CPUs are also stupidly cheap. Assuming Ivy Bridge era (yes, this is old, yes, it's still plenty fast for nearly any web app), even the fastest available (E5-2697v2) is $50 for a matched pair.
I don't say all of this just guessing; I run 3x Dell R620s along with 2x Supermicros in my homelab. My uptime for services is better than most places I've worked at (of course, I'm the only one doing work, I get that). They run 24/7/365, and in the ~5 years or so I've had these, the only trouble the Dells have given me is one bad PSU (each server has redundant PSUs, so no big deal), and a couple of bad sticks of RAM. One Supermicro has been slightly less reliable but to be fair, a. it has a hodgepodge of parts b. I modded its BIOS to allow NVMe booting, so it's not entirely SM's fault.
EDIT: re: backups in your other comment, run ZFS as your filesystem (for a variety of reasons), periodically snapshot, and then send those off-site to any number of block storage providers. Keep the last few days, with increasing granularity as you approach today, on the servers as well. If you need to roll back, it's incredibly fast to do so.