I have gotten the Gambit seal and I'm also just two guns away from the Destinations seal. They're both not too bad, but you'll commiting to two very different experiences here.
Let's do some math real quick. For Gambit you'll have to get 45.000 infamy points, which is that mode's experience point system. The infamy points you get vary for the different ranks you obtain, but as a general rule of thumb, it's somewhere between 130 and 250 points per win, depending on your winstreak. Losses award 30 points. Once you hit 12.000 infamy, you stop gaining infamy for losses all together, so you have to win or you will lose a lot of time. Since you need 2 out of 3 rounds won, matches can take 15 to almost 30 minutes. Don't underestimate that. So, if you were to win every single game, that would mean 45.000 / 250 = 180 matches wins. That is if you never lose your streak. A match takes 15 minutes at the very least, plus loading times that's a maximum of 3 wins per hour. That means it takes a minimum of 60 hours for that seal, even if you know what you're doing and even if you're playing with a team. Anyone who says you can do this or should do this on your own doesn't know what they're talking about. I've easily played this mode over 100 hours and it is infuriating to play without communicating. You can't carry people in this mode. If people don't know what they're doing, you can't win. Unless the other team has even less of a clue what they're doing. Play with a team or get the Destinations seal instead.
For everyone who wants easy gambit wins, here's what to do:
1. Communicate with your teammates. Don't be a dick, try figuring out what the team as a whole can do to improve without blaming someone needlessly. If your team falls apart, so will your winstreaks.
2. Run Ikelos shotgun and Sleeper Simulant. Ikelos shotgun will melt any blocker in seconds, even Ogres. By seconds I literally mean seconds. Maybe 3. Use your primary against normal enemies and your Sleeper against other players.
3. Don't horde motes needlessly. Ogres are easier to kill than small Phalanxes because of their shields. If you are on the verge of a threshold like 24 motes, it's okay to bank just 1 to be able to invade. You wanna deny the other team their motes. If you destroy them in the first round, a lot of people will quit out, making it even easier to win the second round.
4. Run 2 Hunters and 1 Titan. 4th player doesn't matter. Kill the two witches once the boss spawns, get Primeevil slayer to x2, put a melting point on the boss and use your blade barrages. This will instakill your primeevil. There are variations of this. You can have 4 hunters and let them use their R1 blades instead of melting point. It doesn't matter. The health bar will drop within 3 seconds, literally. Once you get your boss up, the other team can invade, which they usually do instantly. While that happens, take down your witches as a team using Sleepers. Even of the invader kills your entire team, you will take down the boss before they can invade again.
This is the reason why you wanna play as a team. I played a lot with randoms and I reset my rank twice like that, but this is way way WAY easier as a team. I would estimate that I played 50 matches with a communicating group of 4. I lost maybe about 2 of those with the strategy I just told you about.
While going for this triumph seal, I got the hidden servitor boss to spawn 7 times. Killed him 5 out of those. Only one of those times was me playing with a group. Once they patch the spawn rate, this will be very easy to do since the spawnrate atcweek 3 of the curse cycle will be the same as for all other bosses (bringing it up to 25% or something like that).
On the other hand, let's talk about the Destinations seal. This can be done without a group, for the most part. It's mostly just doing every Lost sector once, which is easy and quick, then doing Heroic adventures on every planet, which are cycling weekly, which is therefore timegated but not at all hard. Whats more annoying is that you have to complete 3 collections badges. The one for the Red War is easiest. Just buy all 4 sets from the vendors of the 4 base game planets. Get the materials from spider. His resources change once per day, buy them at a rate of 1 legendary shard for 5 resources of x. The more grindy stuff are the Mercury verse weapons, which are timegated too. Takes 3 weeks to get them. Grindy, but easy. You need all 3 Escelation Protocol weapons, which can only be dropped from the bosses at lvl 7. You also need all 5 gear pieces from the chest at lvl 7, and you can only get 1 per week. For this alone, starting out fresh, it will take a minimum of 5 weeks to get this seal. Then you need the Strange Terrain strike as a nightfall to get the braytech osprey rocketlauncher. A friend of mine got it twice within 4 runs, I still don't have it after 40+ runs. Also you need the weapons from sleeper nodes, which are 4. I've opened close to 40 nodes and am still missing a gun.
So, if you wanna be done quick and if you're starting fresh and if you got 3 guys to play with, I would argue Gambit is the mode to grind out here. If you wanna go slow and don't mind grinding solo, then I guess Destinations is the one to choose.
Dreaming City also shouldn't be underestimated since there are dragon eggs to shoot, which are also on a 5 or 6 week rotation, plus the drop rates for the ghost and sparrow and ship seem very low. I also suspect (though I'm not sure), that you need the 1000 voices raid exotic for the collections badge, which, if true, could also be very grindy. I've opened 8 raid chests so far and haven't gotten it yet.