The answer is, "It depends!" Your question also alludes to this. It also depends on perspective.
As a tester/quality engineer, yes, it can be important! Even as a developer, it's important to be aware of any messaging occurring in the browser console. I personally always have the browser console open when I am testing. This allows me to see errors/warnings when I find a bug, which will help in root cause analysis. I take this information and add it to the bug report. I also look at the network tab in DevTools for other errors.
Some devs I've worked with are surprised I do this and are usually appreciative of this. (The surprise is there if people assume all testers are only performing black-box testing, which isn't the case.)
While I don't have a number, there are a lot of people that do testing as an entryway into tech and don't get the training needed to consider looking at the browser console or how to utilize the DevTools. I recommend people learn this! There is so much information DevTools provides. So if you have anyone on your team doing testing work, and they are not familiar DevTools, I'd encourage you to get them some training here.
There can also be a lot of noise in the browser console from browser extensions. One area of improvement can be info logs. Some developers leave this here and they can be removed! The less noise the better as it'll make it easier to identify real problems! Some warnings can be left. But I'd argue that errors should be removed. Of course, any log info should be triaged to determine severity/priority/scope. Does any of this info cause any legal liability? Any data integrity issues? Will this information cause a hacker or nefarious actor to gain information for exploits? This is where security testing can help! These are only a few concerns I'd have if errors are not cleaned up.
It sounds like the developers on your team are arguing against "extra work" or only considering if they cause functionality issues. As noted, there are lots of non-functional reasons to fix them.
they say some of these errors cannot be removed from the console hmmm, I'd say this isn't really true. Errors can be removed, they just might not know how to remove them (an opportunity for training). Or, could be an error coming from a 3rd party dependency; in that case, they are right: it's not their responsibility to fix those. However, ensuring dependencies are up-to-date with the latest versions can help with this and is their responsibility. I'd recommend using tools like Synk or Renovate (other SAST tools) to help with dependency management and security scanning. Some of these tools can automate the upgrades of dependencies.
Another area of interest is "depreciation warnings." These often come about because the JavaScript, HTML, CSS features being used are older and will become deprecated soon! In order to meet modern standards, it takes awareness of seeing these in the browser console to make a recommendation to correct them.
From a user perspective, this may not matter. I doubt most users, especially non-technical users, have any knowledge of the browser console/DevTools, which also sounds like the argument your developers are pursuing. Users would only be aware of problems if it stops them from using the software.
I'd keep pursuing why they don't want to fix them to see what's really true. Using 5 Whys concept can help to determine a reason.
I agree that leaving console logs can cause doubt (does this company care about quality?). It can look sloppy and unprofessional.
