Multiple switching LED drivers on single PCB
17 different switchers might sound like a lot, but it is "just" 24 dB more emissions than having only one. Well designed switcher is easily 30 dB from the EMC limits, so there shouldn't really be a problem. Of course you need to design them well, but it's still not as difficult task as you might think.
Without doing more math, to me the EMC trouble seems easier to solve than the heat trouble you'll have if using linear drivers.
Using DCDC buck converters is definitely the way to go. You can find inexpensive types in SOT-23-6.
This one might be just the ticket: https://www.diodes.com/assets/Datasheets/AP63200-AP63201-AP63203-AP63205.pdf - it uses spread-spectrum to reduce peak EMI and controls drive to reduce ringing at the inductor.
Use shielded inductors for the DCDCs.
Consider a common-mode filter between the regulators and the LED array. You could do something as simple as split the power supplies onto a separate board and run its cables to the LED array through a common ferrite core.
If you use the separate-wires approach, you can use the low-side of the LED string for current sensing for your LEDs.