So I'm wondering what's the best way to ensure they are received by the user? Right now I'm thinking of using OVH SMTP (ssl0.ovh.net) but I'm concerned the emails might end up being blocked. Any suggestion?
Setup SPF and DKIM, check it's all correctly setup. Be aware that if you're using SPF and you are sending emails from multiple servers (e.g. business email account like outlook365 and a transactional email service) it has to allow all the servers.
"If the email hasn't arrived within an hour or so, check your spam bin."
It can be mitigated by "doing things properly" but you can never be sure. Make sure your email is contentful and meaningful.
Anecdata: I use sendmail pretty much directly from my shared-hosting service and things very rarely go missing.