Apple Mail Client Won't Send SSL Email Via SMTP

My Apple mail client suddenly stopped allowing me to send email from my domain six days ago, even though all the settings remained unchanged, the same as they've been for several years -- incoming via IMAP on port 993 and outgoing on port 465 with SSL via SMTP.

I use the Apple Mail client to send and receive email from five different accounts on five different domains (including my MAC.COM account). I can still send and receive using the four other accounts, including a couple gmail accounts and two other personal domains. The two other personal domains also happen to be hosted by the same hosting service as the problem email, albeit on two different servers. Regardless, the settings are all the same in the Apple mail client -- IMAP and SMTP on SSL, port 993 in and port 465 out. Those all function fine. It's only the one problem domain that won't let me send via the Mail Client (it continues to allow me to pull messages off the server for the problem domain).

I contacted my domain's hosting service for help and, needless to say, they sort of wasted six days of my time making me erase and rebuild the universe, all to no avail. I've since erased all the accounts, all the outgoing servers, all the logs and prefs and every other thing from the Apple Mail client and yet, the problem persist, even when the only emails being accessed by the Mail client is from the one problem domain.

This afternoon, the hosting service told me that the reason for all of these problems is this:

***My Apple Mail client is so outdated that it's no longer capable of supporting the SSL requirements of my domain's server.***

Good theory, but here's a wrinkle: What about my two other domains hosted by this same hosting service, each on a different server, for which I continue to be able to both send and receive email via the Apple Mail client, both with all the same settings as the problem account? I suppose it's possible that the hosting service hasn't updated the security levels on all it's servers yet and that it would only be a matter of time before those domains' email accounts suffered the same woes. Nonetheless, I thought this detail worth mentioning.

So, here's the question: Does this outdated-software/outdated-OS/outdated-hardware theory hold water?

My OS is Yosemite (10.10.5).

My computer is a 27-inch iMac desktop mid 2011.

My Apple Mail client is "version 8.2 (2104)."

Here's a snippet from the server logs (my IP address has been replaced with XX.XX.XX.XX):

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2018-06-28 13:28:43 TLS error on connection from [XX.XX.XX.XX]:38601 (SSL_accept): error:140760FC:SSL routines:SSL23_GET_CLIENT_HELLO:unknown protocol

2018-06-28 13:28:43 SMTP connection from [XX.XX.XX.XX]:44614 (TCP/IP connection count = 43)

2018-06-28 13:29:08 TLS error on connection from [XX.XX.XX.XX]:44614 (SSL_accept): error:00000000:lib(0):func(0):reason(0)

2018-06-28 13:38:48 SMTP connection from [XX.XX.XX.XX]:35880 (TCP/IP connection count = 37)

2018-06-28 13:38:48 TLS error on connection from [XX.XX.XX.XX]:35880 (SSL_accept): error:140760FC:SSL routines:SSL23_GET_CLIENT_HELLO:unknown protocol

2018-06-28 13:38:48 SMTP connection from [XX.XX.XX.XX]:35093 (TCP/IP connection count = 37)

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Thank you in advance for all your helpful guidance. I appreciate the forum contributors very much.