Depends on the ISP, don't you think? I work for an ISP that sells e-mail as a separate service. If you're in the service area of a suitable provider, then you can buy e-mail service without subscribing to any network connection service, anyways, so it doesn't matter if you switch ISPs.
We're not "giving e-mail service away"; we are not Gmail. If you want a free/gratis mail service with all their disadvantages (and advantages) such as larger attachments and more theoretically allowed disk space with the disadvantages of lack of professional on-call management and no phone number to call which a competent human will answer, or no option for hands-on assistance from a human being if something major goes wrong with your service or account, or you get stuck, then go over there to one of the major search engines for free webmail by all means.
E-mail is a complex application which is totally separate from network connectivity and requires application-specific management for reliable operation. Why should the two services ever be treated as if they were part of the same? They're totally different services.
If reliable e-mail access and delivery is of the umpost importance to you, then you should self-host, or use a paid account with an ISP or hosting provider.
Because it's definitely a better idea than using a free Hotmail account.
There is also a totally different set of skills and experience required from professionals implementing and maintaining e-mail systems, from maintaining a network.
There's no reason you should not be able to switch ISPs but keep your e-mail and DNS hosting, if you want.
Of course you still have to pay the hosting bill to some provider, and it's probably somewhere between $120 and $150 per mailbox.
If you purchased your own domain name and hosted e-mail under that domain, there's no reason you shouldn't be able to take e-mail service to any willing host.