If you make open source software, the solution is simply that you should not distribute your software [1]. Let distro maintainers do it, and suddenly it's a lot easier for you.
If you make proprietary software, then it's not exactly the problem of the Linux distros anymore so sure, feel free to link statically. I personally think you should do a mix of both: libraries like openssl you want to link dynamically from the system. Now some obscure dependency you may have that breaks compatibility randomly and is generally not distributed by major distros, probably you want to link them statically. But you should know that you are then responsible for them and the security issues they may bring (i.e. you should monitor them and update them regularly).
[1]: https://drewdevault.com/2019/12/09/Developers-shouldnt-distr...
I’m happy for someone to update the version of my dependency if it has security patches. But the rest? A total nightmare. I’ll take cargo / npm / swift / etc style versioning every day.