I have used fastmail for all my business accounts for years. I have never had problems with deliverability. But I did warm the accounts using best practices recommended online to start.
Basically, you have to take a set of ritual steps to "warm" various email resources (such as domains, addresses, and IPs) to be flagged as "known" and "legitimate". For example, IP addresses: you should send warming emails to a valid address that you control for a few weeks to just show up on lists without having spam marks against you.