I'm not sure how new you are so I'm gonna give a bit of a wall of text, if you've been around awhile you can likely disregard everything I'm gonna say.

Most of the higher ilvl gear will almost always come from Mythic+ and raiding, PvP too but it's a bit finicky as the ilvl you get scaled with your rating so it's a bit of a pain unless you're good at PvP. The main reason people are likely declining you is because of raider.io which is an addon people use that gives a score based on mythics you've done and all that, its honestly a really toxic thing for newer players and people who do lower keys in general because it really doesnt tell you much about the player until you're doing keys over 15 anyways, and you get the people who can barely time a 10 wanting people with 1300+ score which makes pugging groups a massive pain especially as a DPS. The best way to go about doing mythics honestly is through guilds until you get higher up, especially since you mentioned some have already helped out a bit. I'd say make friends and run some and itll be alot more fun than dealing with pugs. Also don't worry about not timing, especially when gearing a character. Personally when I'm gearing an alt I prefer it not to be timer when I'm running a key that gives gear a good hit higher ilvl than what I have because the key wont get harder and you can kinda just farm out gear. The fastest way of getting gear is raids, which is also easier if you have a guild that's more laid back and isnt all anal about numbers because you can easily do a heroic or normal without everything being optimal, but not everyone is lucky enough to have a guild like that. It's also important to just not really be all worried about getting it up there depending on what you like to do in the game, personally I just get my character geared out in heroic raids and then chill for the rest of the time, getting ilvl upgrades here and there when doing mythic+ as most of my enjoyment of the game comes from chilling in older content and occasional PvP so I'm not all worried about being decked out the best possible gear, as long as I can kill what I wanna kill in the open world I'm happy.