Free email services have drawbacks you should be aware of before relying on them
If you send an email you implicitly have an expectation that it will be received by the recipient timeously or that you will be made aware (also timeously) in the event of there being a problem.
Most free Email services are a numbers game. The marginal cost per client is negligible so they work. This means (as with any online service) that humans need to be kept to a minium. Unfortunately a free email service provider has probably cut them out altogether.
This article looks at why you should think seriously about using free email services (for businiess or pleasure).
The most obvious argument is the "no such thing as a free lunch" [https://en.wikipedia.org/wiki/There_ain't_no_such_thing_as_a_free_lunch], but since CIS offers a paid for service this is a factually correct, but self serving argument.
The problem is not around access to "your" stored email. Gmail, Yahoo! and Office365 all have reasonable track records at ensuring email you have stored on their system is accessible via the Web, POP, IMAP etc.
The problem is with delivery. Email is different to other components of the internet in that it is human to human, via store and forward. Other internet serives are system to system or system to human, instantaneous. Email is unique in this respect.
Reliable access to your stored email is good, but what about if your emails are not received by the human on the other side? (or if you are the human on the other side not receiving the email).
Any service has costs. The NHS is free at the point of consumption, but there are still costs.
Similarly free Email service providers need to meet the costs of providing their service. There are two sides to this - funding the costs and keeping costs low.
You're using a free service that has costs to the provider. They're almost certainly mining your data, either overtly (you see targeted ads) or covertly (your contacts details are being passed on along with keywords).
You're just a number. It is in the provider's best interest (through the need for cost reduction) to ensure you are unaware of non-delivery. It is also in the provider's best interest (again for cost reduction) to ensure the sender is unaware of non-delivery. The less support it has to do the lower it's human cost.
What is the cost to you if your email is just silently "dropped"?
Free shared email systems will be abused by senders of Spam / UCE, causing reputational issues for the email provider. Your emails will suffer from a bad reputation score. This scoring is automated and (again in keeping the costs low) it takes a signification issue for the free provider to (a) notice the problem and (b) withdraw services from an abusive client.
There are some interesting articles showing some of the issues created by Gmail.