It would be even more of a pain if your email services suffered interruption if the host ceased trading as it would be a very unplanned move. And if you have your domain registered through them, even worse.
I have migrated a number of companies to Office 365 and as has been said, with the right planning and preparation there should be no mail loss and little inconvenience.
If you were getting amazing support I'd say don't worry so much but as it sounds like they aren't very helpful I'd ensure your mail is secure and your website recoverable.
In terms of the problem it's an odd one. If you don't reboot your router when the connection fails, does it ever start working or are you blocked forever? And I presume you are blocked from your mobile on wifi, but I see your mobile on mobile data works. Do.you have any devices or software that you may have forgotten about which could be doing failed logins; a tablet or different mail software (eg/ Windows Mail when you usually use Outlook) could be causing invalid logins? Does the host's answer just relate to them not seeing any failed FTP logins, or were they implying no failed logins for any service?