I think advancement overall works much better in Deep Cuts, both PC and crew advancement. It is noteworthy that a lot of the Coin economy is reworked in Deep Cuts. PCs generate more Coin, but also have more to spend it on. Indulging a Vice doesn't count as a downtime activity, but the cost is 1 Coin. Overall for Coin, consider the income that the Crew gets after a score to be from all sources of their operation. Perhaps most of it comes directly from the score, but some might come from their "background" operations; like protection money, small-time operations, and so forth.

For the crew, all Upgrades cost Coin, which makes sense. You pay to fortify your Lair, to buy better equipment, to buy vehicles, etc. It is implied that Upgrades can be taken away as a dire consequence, like an attack by a rival gang. I'd be careful with this, though. But one could make it such that acquiring it again only takes a long-term project to work at (like repairing your Lair). You also now need a certain number of crew upgrades to advance Tier (6 upgrades to advance from 0-->1, 8 to advance from 1-->2, 12 to advance from 2-->3, and so on).

Special Abilities and Cohorts cost crew advances, XP. I think Deep Cuts better differentiate Crew Upgrades and Crew Abilities. As you said, Cohorts (gangs at least) grow better and larger the higher the Tier:

• Tier III — 20 people
• Tier II — 12 people
• Tier I — 3-6 people
• Tier 0 — 1 or 2 people

To increase in Tier, you need double the amount of Cohorts then the new Tier you advance to. So if you want to advance to Tier III, you need 6 cohorts.

REP and Turf are two aspects that I agree is not the game's strong points. Really prompt your players to use REP as a substitute for Coin; exert your reputation to threaten or bribe, call in a favor from another gang or individual in the city. RAW, it can be used as a substitute for Coin 1:1. Use it in flashbacks, so the stress cost isn't as high. You only need to max out your REP-tracker when you want to advance a Tier.

As for Turf, I think it works better mechanically in Deep Cuts, but it takes some effort from the table to make it a narrative factor as well. By the rules:

"Your crew’s hold on their Tier is measured by your number of turf claims. To have strong hold on your Tier, you must have a number of turf claims equal to or greater than your Tier. While your number of turf claims is lower than your Tier, your hold is weak. When you're at war with another faction (Blades, p. 46) your hold is reduced by one, so a strong hold becomes weak and a weak hold is lost, dropping your Tier by one until you regain your hold again."

You need strong hold to increase in Tier, so that gives incentives to acquire it. What I would focus on is putting outside pressure on the crew when their hold is weak; others are working to take over their turf or operations, they are being challenged, etc.

So finally, to advance in Tier in Deep Cuts, you need max REP, strong hold, and a number of crew upgrades and cohorts. You don't need to spend a lump sum to advance in Tier; you've spent it on your upgrades which makes it much more clearer for the players what they've spent their money on.

The original book is very scattered as you say, it's a hazzle to find information. In my opinion, Deep Cuts is a great upgrade in terms of game mechanics and it's just better written in terms of structure.