GoDaddy is notorious for terrible service and NameCheap has started doing some shady stuff too lately. Luckily there are other decent registrars out there. I can recommend Netim.com or INWX.de in the EU – they also provide EU-specific TLDs which American registrars don’t.
If you need more than one mailbox you can’t beat the offers from providers like PurelyMail/MXRoute/Migadu, where you pay for the storage instead of per-mailbox. I’m using Migadu because, again, they work under EU/Swiss privacy laws.
You do not need to spin up your own mail service and should not. Email and DNS hosting are the most abuse-prone and easy to mess up services; always go to an established provider for these.
Are there concerns tying my accounts to a service that might go under or are some “too big to fail”?
Look into their history. Generally speaking a provider that’s been around for a decade or more probably won’t dissapear overnight; they probably have a sustainable income model and have been around the block.
That being said nothing saves even long-established providers from being acquired. This happened for example to a French service (Gandi) with over 20 years of history.
The only answer to that is to pick providers that don’t lock you into proprietary technologies and offer standard services like IMAP, and also to keep your domain+DNS and your email providers separate. This way if the email service starts hiking prices or does anything funny you can copy your email, switch your domain(s), and be with another provider the very next day.