Li-Ion batteries are problematic because they do not deliver the currents needed for FPV flying. Take a look at this diagram: https://batteryuniversity.com/learn/article/discharge_characteristics_li
It tells you that the specific 3000mAh cell tested there loses around 30% of its capacity when discharged at 2C, which corresponds to around 6A, which is not much in terms of FPV flying. If you draw more current, the loss will be even greater. This is also true for LiPo batteries, but they're made for tens of amps of discharge, and the relevant effect will only manifest itself noticeably at discharge rates of, say, 30C.
Saying that, you can discharge Li-Ion to much lower voltages than LiPo, typically to slightly below 3V instead of the typical 3.5V for Lipos (assuming you don't want to damage the cells or accept capacity loss). So you'd have to adjust your copter warnings/beepers accordingly. Data sheets and available discharge plots at https://lygte-info.dk and https://batteryuniversity.com mostly suggest to not go below 2.5V - 2.7V, if the lifetime of the cell is to be achieved.
Then weight is an issue. A typical 4S 1800 mAh LiPo weighs in around 200g, so you get 9 mAh/g. A modern high-current Li-Ion 18650 cell such as the Sony US18650VTC5A with 2600 mAh weighs 48g, so you get 2600 mAh / (4 * 48g) = 13.5 mAh/g. Now if you take into account a small capacity loss because you're discharging the Li-Ion at 10A ~ 4C (flying very carefully!) you'd probably end up with 2200 mAh or 11.5 mAh / g.
A potential way out of this is, of course, to use more 18650 cells to make a 4S2p pack, which would half the discharge rate per cell and therefore give you not only more capacity in total (2 x 2600 mAh = 5200 mAh), but you'd also get closer to the nominal cell capacity because the 10A you're discharging at are ~ 2C and not ~4C as for a 4S1p pack. But then again you'd carry around a 8 x 48g = 384g battery, which would probably feel noticeably more sluggish than a 200g battery.
The Samsung INR 21700-40T (this is not a 18650, but it's slightly larger!) has a capacity of 4000 mAh, can deliver 40A and weighs 70g, resulting in 4000 mAh / 280g = 14.3 mAh / g when assembled in a 4S1p pack. Discharging this at 20A to 3V results in a capacity of 3300 mAh, or 11.8 mAh/g. This might be a viable option for a long-range copter.
So while Li-Ion batteries probably are the next best option compared to LiPo, they can't compete in racing or acrobatics, but they may outperform LiPos in long-range situations.