Yeah, most of the PUG leaders will only accept 5.7k GS + achievement, because it’s easier for them that way. They don’t know if you actually know what you’re doing, and gear can compensate for some mistakes. Also, higher GS usually means the player has already been in some raids, or at least is an adult who can buy gear—unlike some kids.
But sometimes, you can catch someone who accepts lower GS, mainly in chill alt runs or late pre-reset raids (Monday, Tuesday).
Also if you have time to play consistently, best option is joining good guild that is progressing. They will accept you with a lot lower gs if they know you are a good player or wont mind teaching and boosting you a little, because its investment into the future for them. But sometimes people play irregularly and cant or dont want guild, ok.
If noone accepts you into PUGs and you are without the guild, then just start your own raid. I know it’s easier said than done, and it’s definitely a lot of work, but you can start ICC even with a 5k GS character. Accept a mix of high-low GS players (you’ll need at least one high GS tank and one high GS healer) and go for ICC 10 Normal. Get your Emblems of Frost and buy T10 gear right away with the reputation for BiS 277 ilvl ring as side dish. Just don’t reserve anything, be kind, explain everything, and run a chill raid that helps you and others get out of the low GS trap.
I did exactly that with my feral druid alt, and we managed to kill 8/12 on my first try—with a 4k GS mage in the group. (His friend, a 6k GS ppal/retri, was part of the deal—I accepted them as a package. The mage never messed anything up, and we geared him through the run, so his average DPS was over 5k by the end.)
If you’re new to raid leading, I’d recommend first trying some weekly Q raids like Naxx, ICC rep farm or VoA (in rep farm and VoA you can ress whatever you want). After a few successful smaller raids, go ahead and try ICC.
And if you dont have the knowladge, set up and/or skill to play and lead the raid like that, there is still hope for you. You can get some higher gs PvP gear to boost you into some first raids. You can get basicaly free honors by doing Wintergrasp weekly Q, or you can actually join WS battle, or random BG. With honor you can buy 264 ilvl offpieces. For some classes (feral, war and so on) PvP gear is just too big of drop in performance, but some classes like DK perform okay with it, even trough some raid leaders still dont like to see it.
Sidenote on why some classes shouldnt use PvP gear and for some its kind of okay or even advised? Well feral, war, hunter and classes like that get huge boost from all stats like ARP, while unholy DK or retry paladin doesnt benefit from ARP too much, and a lot of PvE gear they can get is ARP gear. If you replace 232 ilvl PvE ARP gear with 251 or 264 PvP gear you will actually do more dmg thanks to higher crit and STR, even trough resil is wasted stat in PvE for DPS it will still end up better. And for tanks like blood DK are PvP items even better. They offer a lot of stamina, sometimes even more then PvE tank items of similar ilvl and they actually benefit from resilience. If you are 5k gs half-*** tank and you are not defense capped (540 deff), you can look how much crit hit reduction you get from your deffense and get to your desired 6 % crit immunity using some resilience which also reduces crit hit chance. You will maybe get 4 % from you PvE tank gear trough deff stat, and 2 % from few high ilvl PvP items that can replace your lowest ilvl offpieces.
Now when we know PvP gear can be sometimes kind of okay in PvE performancewise and also leaders sometimes wont check for it in lower PvE raid content like ToC or ICC 10 normal i may add haw you can efficiently get 232 (honor) or even 251(arena point) PvP set items (head and chest) to greatly boost your overall gs. If you have partner, or second account you can join 2s arena every week with 4 chars in team and play 5 games on 2 of them and second 5 games on the other 2. In that way you can get your weekly arena points (and 251 ilvl set pieces by extension) for up to 4 chars at the time with nearly no work, because you just need 10 games of your team played and to be in more then 30 % of them. You dont need to win any of them, just go for 10 games and after a week you will get 261 AP on every char. So after 4 weeks worth of AP using that method and few random BGs you can have your 251 ilvl PVP set chest and head both offering 430 gs.
Another sidenote, why do you want to buy PvP head and chest instead of other set items? Because you dont want legs and hands that can be obtained (in superior ilvl) from VoA 10 and 25 in both PvE and PvP versions, while shoulders offer smaller gs boost. Shoulder also can be bught as T9 or T10 with EoT or EoF cheaper then head or shoulders and casters can get their 264 ilvl shoulders as BoE for few golds from AH.
So i would say the 5,7k gs + achiev requeremnt to join any PUG raid is here, thats sad reality, but there definetely are ways to progress without spedning coins and by extenision real money. Sure 500 gs boost by buying 284 ilvl weapon for 23 coins is also option, and its a good option if you have money and only limited time for games. But if you dont have money to spare or are against any pay to win, there are ways to progress, even to progress efficiently and there is actually lot of them.