Should you use your web host’s DNS service? If your web hosting provider is providing free DNS service. You may consider using it if that is what you need. Most web hosting providers and most domain registrars provide complimentary DNS service with their respective services. If your website is a local community website or you don’t value it always being available (not critical) then it should be alright.
If you website is an ecommerce website or a business website (critical) and the site has to always be accessible. You may consider having your DNS hosted by a managed DNS provider. With a managed DNS provider, they would handle the DNS requests while you worry about your website. If you hosted your DNS records with a managed DNS provider and your website is down. Depending on the managed DNS provider you choose and the the plan your respective DNS provider has offered. You may be able to change your site’s IP address to another web host when your current web host is suffering downtime for any reason.
Critical sites = go with managed DNS service provider
Hobby/Non-critical sites = go with web host DNS or domain registar DNS service