World of warcraft gear guide
My suggestion would be to either look a raiding guild / cross realm raiding group for getting into heroic raids and/or to find a regular group for mythic+ that at least maxes out one key each week for the notably higher quality weekly mythic+ chest drop.
You're also in luck with 7.1 patch adding Karazhan dungeon. That place has 860 baseline items and the last boss to my understanding can drop 875 pieces.
As for pugging mythic+, it's kinda expected to have slightly inflated requirements for difficulties. I have a group of maybe around average 865 ilvl decent players, and we usually cap around +8 mythic runs for weekly chest (we don't run them too much, but then again only very few people actually spend time on these to be really good). Throw that same situation into a pug, and 865's that don't know each others will have quite a tough time. Not saying it's not doable, but it does have certain problems.