Actually, if your only aim is to make orchestration rolls, you DO NOT need to level everything to 60. But you will still need the following:
(1) Reach ALC 60. Every time you level up, lower level recipes will become significantly easier to complete. Thus, by reaching ALC 60, it'll allow you to complete orchestrion rolls of recipe level 50 to 56 pretty easily. Another reason of reaching ALC 60, is because many orchestrion rolls are actually locked in Master Recipes (ARR Master Recipe, and Heavensward Master Recipe 1 & 2). To obtain these master recipe, a lvl 60 class is necessary.
(2) Obtain critical cross class skills that has to do with increasing "progress" for item completion. There are many cross class skills that will benefit you on progress. Piece By Piece (PBP) is a MUST. You can't handle high level orchestrion rolls without it. So get ARM level 50. I don't think anyone would disagree with this. If you have ARM 50, you would have gotten Rapid Syn at ARM 15 anyway, and it's another skill that is great for progress. However, Rapid Syn sucks on fidelity, so you will need CUL 37 for Steady Hand II. Other things to consider include WVR 50, which gives Careful Syn II (100% in fidelity and 120% in progress comparing to Basic Syn) and BSM50, which gives Ingenuity II (wonderful to empower any "synthesis" skills like Careful Syn II or Rapid Syn). CUL 54 gives Muscle Memory, which is basically a PBP that can only be used once, but it's 100% fidelity and cost only 6 CP. GSM 54 gives Maker's Mark and GSM 40 gives Flawless Syn. These two skills are also great for progress. But for you, I don't think it's necessary. ALC itself also have Name of Water and Brand of Water. Pairing these two up also gives very strong progress as well.
However, there is a catch here. To obtain ARR Master Recipe, you actually need to turn in some HQ items (lvl 50 ARR 2 star stuff). Back in the old days, we will need "ALL classes 50" to handle it. Now, you can simply BUY the HQ items on the MB (since they're really cheap now), or you can just overpower them using a level 60 ALC. To obtain Heavensward Master Recipe 1 & 2, you will need to obtain Blue Scrips. And to obtain that, you will have to deal with quality in items by making collectables. Luckily, you can turn in recipe level 56 items for Blue Scrips, so I think you can still overpower it with a level 60 ALC, but it really wouldn't hurt to try to obtain some cross class skills to assist this process. Thus, I strongly recommend you at least having all classes to level 15 first. This is extremely easy to obtain, yet the power from all those different lvl 15 cross class skills is tremendous.
Keep in mind though, leveling ALC to 60 without lvl 50 cross class skills from multiple classes will make life VERY difficult. The "All lvl 15" I recommended above will help A LOT, but still not as good as "All 50". Because it will be very difficult to HQ any recipes on your crafting log that are of the same level as your class. You can, however, exploit Moogle Beast Quest and Ixali Beast Quest to obtain EXP every day. You can also push levels using leves by submitting NQ items, but you will need TWICE the amount of leves and TWICE the amount of materials (and gil) to do the same job as submitting HQ items.
Oh, and don't forget, don't mess with melding materia. You won't need it unless you're at level 60. If you really want, you can meld some cheap CP materia to help. But please don't burn all your gil.
Don't feel stupid at all. Crafting can be a complicated game. And despite yourself having a level 51 ALC, most of your other classes are below level 15. So you have barely entered the crafting world yet. To be frank, a little googling on "FFXIV crafting guide" would probably direct you to some useful information. I do recommend you to read a little bit, and learn a little more about it.
Absolute BS. Without cross class skills, the crafting game would collapse completely.
Agree. The Specialist stuff won't help as much as getting "progress-related cross class skills". However, since Dreggit is only concern about making orchestrion rolls, "quality-related cross class skills" are not critical. This means he does not need to push everything to lvl 60. Having gatherer classes leveled up would help save a lot of money though.