Let's give some consideration to the types of shoes (or none) that we use for speed training. There are a few general principles to apply to shoe selection with regard to speed, bodyweight, and terrain.
For the most part, the faster (and shorter) you go, the less substantial of a shoe you need.
Conversely, the heavier you are, you may prefer a shoe of slightly more substance.
There are all sorts of terms we can use to categorize shoes with regard to the amount of support/stability, cushioning, flexibility, and weight they provide. I've been involved in the footwear industry for 40 years and I really enjoy deep-diving on the subject. But that is not necessary for this discussion. Some of the things we do might be in the gym and on rubber flooring or various court surfaces, but if we stick to thinking of the field and track, we can make a few summary statements.
On the field, you don't need or want much cushioning but you do need some traction. Avoid wearing a clunky shoe with a thick midsole because this decreases your "feel" for the surface and has been shown to increase the risk of ankle sprains and knee injuries. You can use a lightweight trail running shoe, a spikeless cross country racing flat, or soccer/football cleats. They all work well depending on your preference. The main thing is not to wear "too much shoe."
On the track, you can certainly get by with a basic road running shoe. This can be in the classification which you prefer, i.e. neutral cusioned versus stability, etc. However, I'd steer you away from the maximalist running shoes with super thick midsoles originally designed for elite marathon road racing. Something in the mid-range of "substance" is ideal. I like to use a lightweight trainer or road racing shoe most of the time because it is protective enough to get me through an entire workout but light and responsive enough to work reasonably well for the really short, fast stuff. That stated, I do have and use (and recommend for anyone who really wants an incredible speed experience) a pair of sprint spikes. They are not expensive and at the level we use them, they'll last you forever. I'll usually do my warmup (or most of it) in the flats (trainers) and then switch to spikes for the main set (again, in most but not all cases). Then it's back to the trainers for the accessory and cooldown work. Even though I'm not that fast anymore, it's sure fun to feel fast wearing those spikes. If you try them you'll have total buy-in.
Lastly, barefoot training on grass, in limited amounts...is spectacular. There is nothing that strengthens your feet, improves your form, and gets you natural grounding and anti-inflammatory benefits like barefoot running. We use it in T2. But we're careful not to overdo it. We progress volume slowly, never get too high, and use the shoes for the hardest stuff.
If you can receive this wisdom, you'll choose your shoes wisely.