I also don't believe it's possible to pick the 'important' areas of code a priori and make only those areas free of technical debt. Doing good work is an attitude, a mind set, and something to practice - if you practice making poor quality code, it's going to affect you across your work, not just in the places that you do it.
I think that, by being professionals and putting aside the 'hack it out' mentality, we can improve not just the code that we're working on, but our mindset towards it. How many codebases have 'that scary part' that adds background anxiety to the people that work on it? Is that something you want to invite into your life on an ongoing basis?
Especially for things as important as integrations into external APIs - sure, they may not change much, but in my experience you really want to make sure they're done right when they're how you deliver your product to customers. Given how error-and-failure-prone many external APIs are, I expect a lot of work there just to robustly handle the other end of your integration being flaky.
I think that quality is something you can't 'bolt on' later. It has to be built in, top to bottom, throughout your organization, code base, and personal skills.