Red Mage and Samurai both have skills tied to quest completion: at Lv60 and Lv70. Although if I remember correctly, the difference is that the red mage quests up to Lv60 are set entirely in ARR territory, while samurai Lv58 requires you to visit Ishgard and western Coerthas (the first 'field area' of Heavensward) so you'll have to wait until then to complete that quest line.
Once you complete the Lv60 quests, they actually do stop giving you quests until you are able to access the next location. Or more exactly, they won't give you the Stormblood set of quests (Lv60-70) until you reach Stormblood in the MSQ, regardless of whether you can access the required locations or not.
For the most part this applies to the ARR jobs as well - the initial job questline takes you up to Lv50, but you can't pick up the Lv50-60 Heavensward-era quests until you reach Heavensward in the story. Red mage and samurai are the only exceptions for this, since they are accessible in ARR and the other option would be to give them nothing at all storywise until you unlock Heavensward.
Access to areas for quest progress does vary a lot from job to job. Some of the Lv50-60 plots take place back in ARR settings and can be cleared immediately. Others require you to wait until you gain access to one of the very last regions unlocked at Lv58 in the MSQ. Likewise for the Lv60-70 quests, which may be immediately accessible in previous zones, or may require you to reach certain new areas, or in at least one instance simply says you can't progress until you have completed quest X in the MSQ.
Even in ARR you'll find occasional blocks, like summoner which - for lore purposes - requires you to have cleared a certain fight before you can progress with the job storyline.
As a general rule, the job quests are written with the expectation that you are playing both job and MSQ at around the same level. If your class level has gotten ahead of the story, you're "out on your own" and the game won't be expecting you to be there yet.
The other thing is that there is no actual issue with you not having access to your Lv60 samurai skill yet - because there's absolutely no content that it's useful for at this point. Any dungeon you participate in will synch down your level to match the dungeon, so you won't be Lv60 and won't have the skill anyway. (It's just a passive "meditate to charge your kenki gauge" skill - and you won't have the gauge in any ARR content either.)
Also if you had directly paid to boost samurai, it would automatically mark the Lv60 quest as done, and give you the ability. It would just be a very inefficient way of spending your job boost instead of using it on a class that starts at Lv1, purely to get an ability that's not all that useful yet.