You mention Q's discussion of when humans were "supposed" to get to the Delta Quadrant, and its worthwhile to point out as well that Q himself had already sent the Enterprise-D to the Delta Quadrant "ahead of time" for their first encounter with the Borg, in "Q Who". Q may be an unreliable narrator, both on account of his nature as a galactic troll of sorts, and given that he has already broken this "schedule". While his statements could provide some indication that Starfleet was likely in the Delta Quadrant by Discovery S3, the continuing growth and reach of the Federation and Voyager's successful return home may be a stronger indication of such. Voyager's return, in particular, would be likely to precipitate further exploration of this area of space.
I don't believe S3 Discovery (~800 years after "Death Wish") made any mention of any specific Starfleet activities or ships in the Delta Quadrant, though it is reasonable to assume that there would've been some sort of Starfleet presence in that quadrant. The Federation's reach appeared to be continuing to grow prior to The Burn, and the only reason to potentially avoid the Delta Quadrant would be The Borg, which could just as easily be a reason to go there in the first place. Most Starfleet vessels in the Delta Quadrant would've likely been destroyed by The Burn (as noted in the comments, those not in warp at the time were unaffected), and any settled colonies would've very likely splintered off when Starfleet did not come for them. As a result, its unlikely any Starfleet presence remains in the Delta Quadrant.
We are told in Discovery that The Burn was a galaxy-wide phenomenon, but given the limited communications that exist in the wake of the loss of most interstellar travel, it is possible that The Burn did not extend across the entire galaxy and only to a certain distance. This is entirely conjecture on my part, but if this were the case, it could mean that some particularly far-flung elements of Starfleet (or other distant civilizations) may have been spared from The Burn. However, we would tend to suspect that such far-flung surviving Starfleet vessels would have returned home and brought such news, on a journey similar to the one we see in Voyager.
TLDR: Yes, there probably was a Starfleet presence in the Delta Quadrant, and it was probably effected by The Burn, but all of that is assumption.
