Do you mean Discovery will literally spore-jump into the Kelvin Timeline, or there will be a reveal that the show was in the Kelvin timeline the whole time? Because I don't think the latter is possible at all. The NCC-1701 is clearly stated to be brand new in 2258 (was even shown to be in construction in 2255), while Discovery already has the NCC-1701 having a long service history. Not to mention the mess jamming Amanda in Discovery and dying in Kelvinverse together and all the other contradictions--Discovery is definitely not Kelvin. The only contradictions with Prime in Discovery are cosmetic, style-based issues (sets, design, etc. obviously not 1960s style), but no actual story or dialogue contradictions.
This always saddens me a lot. I remember seeing them both in person in Sacramento at the 2006 40th anniversary convention, and they sure seemed like good pals. They had a lot of interactions you read about online hugging, chatting, and Nimoy called Shatner his brother. Something really bad must have happened, and Shatner professes ignorance as to what this was. Although Shatner speculates it might be a fight over Nimoy footage used without permission in a documentary, Bob Orci outright says this isn't the case, and that there was another reason, here:
https://trekmovie.com/2016/03/25/review-leonard-my-fifty-year-friendship-with-a-remarkable-man/ (just search the page for 'boborci' and you will find his comment).
I was thinking more along the lines of aesthetic and tonal similarities between Discovery and the Abrams-Lin films. Discovery just 'fits' better with that style. Though you raise an interesting question. Can things be fudged to have Discovery fit within the existing continuity of the Abrams era films? I took the new 2258 Enterprise to just be the new version of that Enterprise, and not the first Federation ship named Enterprise. With that interpretation, it can just be that the Enterprise that shows up at the end of Discovery Season 1 is the version that will be decommissioned or destroyed, making way for the 2258 Enterprise, perhaps named in honor of the fallen starship?
In the Discovery timeline we haven't got to 2258 yet so they can still write in the destruction of Vulcan and Amanda's death. (As I've mentioned in past threads, Amanda's presence muddies the whole Georgiou as human mother/mentor to Burnham anyway. I mean, Amanda seems like a pretty attentive, supportive, and understanding mother so I don't get why Burnham needed another mother figure for the show; a career mentor fine, but the relationship between Georgiou and Burnham is more than just that, it's maternal in many ways). I've seen pictures of the Into Darkness Klingons beside the Discovery Klingons and it looks like the Darkness Klingons are a bridge between the older Prime Klingon depictions and Discovery's depictions. Also, the D4-class ship in Darkness does that as well when it comes to Klingon ships.
If the Picard show deals with the aftermath of the destruction of Romulus, I can see a greater merging of the Prime and Kelvin timelines. With Alex Kurtzman at the helm, and if he's able to do it, I don't see why he wouldn't want to build on what he's already helped create with the Kelvin films.
The idea of Discovery spore jumping into the Kelvinverse is something I hadn't thought before, and I think would be cool.
When it comes to Discovery's contradictions with Prime Trek, the start of the conflict with the Klingons as stated by Spock in Star Trek 6 doesn't quite work with Discovery. Also, the idea that a quadrant-spanning war with the Klingons is never mentioned in TOS or other Treks is also strange-not a canon violation per se, but just seems like something they would have mentioned. Memory Alpha had to alter their entries, because for a long time, the 'first' Klingon War took place during "Errand of Mercy" and the second was on DS9, but now they've had to add this war. I'm sure there are other small things, but for the most part I haven't seen where Discovery has overturned canon, though it has complicated it. I don't have a problem with the Klingon War, though I would've put it about 20 years earlier (like Axanar did) to get it away from potentially complicating anything with the TOS characters. I'm more 'bothered' (too strong a word) by the line that the Klingons have been off the scene-mostly-for 100 years. That didn't feel right to me, it contradicts what Spock said (which would've had the start of the Klingon-Federation conflict as starting around the 2220s), and was an unnecessary line anyway. It didn't serve to do much for Discovery's story to have the Klingons be off the scene. (It's not a contradiction at all, but I do wish Discovery would mention the Sheliak. It would be a nice nod to TNG IMO, and predated the Klingon War by a year).