How much loyalty deficit(I mean the negative per turn modifier that appears to the settler) can newly settling cities handle ? You may answer the question in accordance with different variables such as food surplus, governor use, and whether other things that impact growth and loyalty are used or not, better if you specify them.
+8 if you have a governor available, +1 if you can buy/ chop a monument and +2-3 for rapid pop growth from high food tiles/ food chops. So -10 to -12 is easily survivable.
Disloyalty gives a growth nerf, so the first turns are crucial. If you can buy a monument and chop a pop on turn 1 you can survive a small negative long enough to grow before you hit the 75 loyalty -25% debuff.
If you have a religion there is a -3 or +3 depending on if the city follows the religion. Given the need for rapid growth if you are relying on quick pops have a couple of missionaries present.
Victor with garrison commander in a nearby city is another +4.
Amenities give -6, -3, 0, +3, or +6 but it is problematic to rely on forcing amenities to a given city until national parks come along with their +2 to host city and +1 to four closest cities. Forcing happiness lets you build in the first couple of pop, but bare in mind that population pressure is capped at +/- 20 while the amenities are a separate modifier. There could be niche strategies of keeping a triple NP city tiny and stacking the +6 with other modifiers to eat the max -20.
There are a bunch of cards but, in general, using a precious slot for more than a couple of turns is a very high cost to pay. There are a bunch of civ unique bonuses. Persia with a garrisoned unit +5, a governor +8 (+13), a monument +1 (+14), your religion +3 (+18) and Victor GC nearby +4 (+21) can ignore pop loyalty pressure. With the right cards, governors and sufficient resources other civs can too.
It's a question of how much you want to mould your game around this one mechanic and how much resources you are willing to throw at it.