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hi Rado,
I did have the same problem as yours, but solved it already.
The root of my problem was in me network's firewall, which is very strict as for incoming so for outgiong packets for every local IP which is not in a 'golden list'. By the time I've done my google search and got here, I completely forgot that esxi host IP is not in 'golden llist' and asumed that the problem is in esxi host itself. As soon as I've discovered that problem, I've made an addition of esxi host IP to my firewall golden list - that's all. I believe that your problem is nearly the same.
Needless to say that ALL of my VMs feel themselves well as for gmail so for every other Internet service - simply because their IPs was already added in that list.

So, what was my steps, which lead me to the solution:
1. [root@vcsa:~] nslookup smtp.gmail.com
nslookup: can't resolve 'smtp.gmail.com'
# means DNS services isn't available for esxi host
2. [root@vcsa:~] nc -z <my router with firewall IP> 53
<empty line>
# means my router doesn't want to reply on port 53 to esxi host
3.  [root@vcsa:~] ping <my router with firewall IP>
PING <my router with firewall IP> (<my router with firewall IP>): 56 data bytes

--- <my router with firewall IP> ping statistics ---
3 packets transmitted, 0 packets received, 100% packet loss
# means from my router's view esxi host is just an exile
4. I went to my router console and quickly discovered that all packets from esxi host are being rejected with ticking of a corresponding counter. ESXi host IP was added to 'the list' afterwards and I saw totally different replies to the same requests:

[root@vcsa:~] nc -z <my router with firewall IP> 53
Connection to <my router with firewall IP> 53 port [tcp/domain] succeeded!
[root@vcsa:~] nslookup smtp.gmail.com
Server:   <my router with firewall IP>
Address 1: <my router with firewall IP> myrouter name

Name:      smtp.gmail.com
Address 1: 2a00:1450:4010:c0b::6c lo-in-x6c.1e100.net
Address 2: 209.85.233.109 lr-in-f109.1e100.net
Address 3: 209.85.233.108 lr-in-f108.1e100.net

4. And the last, but the most important command is attempt to connect to my precious the required port:
If you'll see the following:
[root@vcsa:~] nc -4z smtp.gmail.com 465
Connection to smtp.gmail.com 465 port [tcp/smtps] succeeded!
# then XSIBackup will send your mail for sure. Same for TLS port:
[root@vcsa:~] nc -4z smtp.gmail.com 587
Connection to smtp.gmail.com 587 port [tcp/submission] succeeded!