I switched from Dev to SRE at the same company, and within like one week of remembering to update LinkedIn job title, the random recruiter messages switched from Dev to "oh we are looking for someone like you with lots of SRE experience" (having worked in SRE for <3 months).
So yea, it's difficult to get traction for something that isn't already your job title.
This is a failing of the hiring manager. If the recruiter can't tell who is and isn't a good fit, the hiring manager should have corrected the situation or not partnered with the recruiter.
I find that practice disgusting personally and will never do it myself or condone it in others, but it does seem to work.
The LinkedIn random recruiter messages are a different game than resume screening. Most of those recruiters are using the LinkedIn search tools. Searching for keywords like "sre" is essentially useless because so many people do keyword stuffing. So instead, many recruiters will either be given or will ask for a list of companies to try to pull candidates from.