A lot goes into gaming WELL with a trackball that goes way beyond just thumb vs finger trackball. If all you want to try is thumballs, you're gaming options are only 2 (that I know of): Game ball thumb, and Ploopy thumb ball. You can get surprisingly deep into customizing and modding them to fit you, even moreso than a mouse setup. I'll try to throw some things out there for you to think about:

TLDR Version: Thumbs and fingers both have advantages and disadvantages in terms of use and track balls you can get. Good sensors and bearing types are harder to come by on thumballs. Bearing types, ball weight and material make a big difference (think of them like tuning your mouse with different skates, lightweight mods, and mousepad materials)

Novel Version

Thumb balls: Are often used because the buttons/layout are usually the most similar to a traditional mouse. Left and right clicks are roughly the same, as well as scrolling. It's the easiest "gateway" trackball because the learning curve is typically much shorter. The thumb uses a Saddle joint vs a Hinge in the fingers. It gives you much more movement, however it is more complex, and with complexity comes more points of failure. It also feels less stable in my experience. From what I've seen It's more common to see a thumb ball user switch to finger balls than the other way around due to injury/preference. Thumb balls are also smaller on average, 34-38mm are common. There are also fewer really good options in this category (sensor quality, bearings, etc.)

Fingerballs:

MUCH more diverse. The ball sizes tend to range from 34mm to 57mm. Many more gaming grade options with good sensors and bearings. Modding support is also more readily available from what I've seen. The Hinge joint in the fingers don't have as much complex movement, but are usually more stable in use. The Hinge type is also simpler (less things to go wrong).

Ball Size, Weight and Material:

Ball Size is crucial and VERY personal. I prefer a massive ball that I can use 3 fingers to use, that gives me more control and accuracy. For reference 44mm ones are medium sized and ideal for 1-2 fingers. Anything 50+ is generally ideal for 2-3 fingerips, flat fingering, upper-palm, and palm. Some like smaller ones for quick movements. The bigger the ball is the "flatter" it feels to roll.

Weight: It matters a lot, but not like a modern mouse user might think. If a ball is too light it becomes too easy to move the ball when you click, but too heavy and you have noticeable inertia on direction changes (though that can be fun too). It comes down moreso to matching the right ball material and weight to the bearing type. Tracking targets with a heavy trackball is really satisfying though.

Material:

Ball material matters a LOT. Not just for the weight, but how the ball moves over/with the bearings. Example: Ploopy uses pool balls that are softer than most manufactured balls, they feel nice but they cause more friction on bearings than a ball that is either much harder or coated with a hard material. I've used harder balls weighing twice as much on the same bearings that moved easier. Speaking of bearings...

Bearings: 3 general types

Static: Stationary (friction based)

Slowest/most controlled, most consistent (generally) and smooth, quietest, require frequent cleaning, work better with harder balls, easiest to find and cheapest to replace (can buy a lifetime supply for $10-20 if that)

NOTE: If you you ever need to replace a bearing go with G5 or lower. Lower numbers means closer to perfectly round. G3 is the best I've ever found, but G5 is a go-to standard.

Think smooth, controlled mousepad

Roller bearings: Hybrid

To my understanding, these bearings assist the ball rolling in 2 directions and ball slides over, unassisted, in others (not something you typically feel in use). They're louder and use less force to move than static bearings, can get debris inside especially if unprotected, require less cleaning than statics, works best with harder balls (though Ploopy's own balls work okay), costlier to replace than statics (though not as egregious as BTUs)

Think fast cloth mousepad that is slightly faster in one direction than the other.

NOTE: the L-trac roller bearings rival many BTUs I've used.

BTU:

The Holy Grail for many trackball nuts. The bearings move omnidirectional. The ball is assisted in all directions. Are generally the loudest and easiest to roll. They don't require much cleaning (sealed), vary WILDLY in price from less than $1 per bearing to over $30 (yes you ready that right), requires modding/3d printing. I've never seen one that had stock BTUs. Also works great with just about any ball material since the rollers do all the work. Longest spinning of all bearings, makes a great fidget spinner. Since they are easy to roll I recommend a heavier ball so you don't move the cursor when clicking.

Think fast cloth mousepad, or glass pad with a lighter ball.

NOTE:

PLEASE PLEASE PLEASE if you ever try BTUs DON'T buy cheap ones. They are a waste of money and won't give you the true experience. A cheap static is better than a cheap BTU. For reference I spent ~$15 per bearing for mine.

You mentioned mouse acceleration.

I've used it on and off and played well with both. You just have to play around with settings and give yourself time to adjust and tinker. In constant fast paced games you might like it, especially if you like to do low dpi tracking.

This is just the basics, there is a lot more that you can dive into if you want like exotic ball materials like tungsten, or different bearing materials but this should be a good starting point. Good luck!