The most common mess I inherit with a new client is not the website. It is the domain.
They want to move, and it turns out the domain is registered to the designer who built the old site, or to that person's company. Sometimes they have stopped trading. Sometimes they want a fee to release it. Either way the business is stuck, because the domain is the one asset the email, the Google listing, the van livery and every printed card all depend on.
What to insist on, whoever builds the site:
The registrant is your name or your limited company, not the agency. A whois lookup tells you in under a minute.
You hold the registrar login yourself, even if somebody else manages the DNS day to day.
The renewal reminder goes to an address you control, not to whoever built it three years ago.
None of this stops anyone managing your site. It just means you can leave if you want to, which is the whole point.
Source: r/u/hibbaco · by /u/hibbaco