Have dealt with clients in the past like this numerous times. When I first started out I would do whatever it takes to cater to them but since then have learned my lesson to avoid them based on the a-hole scale.
I always give my clients the option...if they want to do their own hosting then so be it but I let them know they won't get any support from me without it being billed hourly. It might save them money but there's no guarantee with it. We typically charge a $50 a month maintenance package to start out using $5 droplets on digital ocean. We deal with mostly local clients so traffic isn't much and in conjunction with cloudflare and Amazon S3 these little servers can handle it. So the $50 gets hosting, domain management, backups, updates and if something happens to the site tech/server related we'll fix it, for example a hack/malware.
On the other hand if they choose there hosting they get none of the above. For some who know how to manage this it won't be a big deal but often times business owners think they can save money paying hosting themselves and before you know it there site gets hacked, they forget to renew etc and then they don't have a backup and start from the beginning.
I always explain to them that what we provide is a service and it's an important one. I think a lot of us can probably agree that websites are under constant attack from spammers/hackers.