I think probabilistic counters are really interesting for this situation, as long as you don't need exact numbers.

  if random.randint(1, 100) == 1:
      db.increment(row)

Then, looking at the database, you can know that the actual count is approximately 100x the number of times the row was incremented.

There's an even more aggressive version that I've seen called a 'Morris Counter' which uses just a few bits to build an order-of-magnitude estimate of the number of events: https://en.wikipedia.org/wiki/Approximate_counting_algorithm

Yes: use a different tool, such as Redis.

Updating a SQL database row can be expensive because of the transaction overhead. Once you increment this counter, the row with the counter remains locked until the transaction is committed. But the actual useful work of incrementing a number is already quicker than selecting a random slot. If the increment can be a simple ADD instruction, without transactional semantics, then this random-slot-based approach is slower than just incrementing it. Especially if you don't care about missing a few increments and therefore don't need a mutex.

For even more scale, don't use Redis but have a shared memory block containing your integer.

For even more scale, you can have one such slot per CPU. Make sure they're in different cache lines, to avoid false sharing. Now you have slots again, but they're per-CPU and so you still avoid all the SQL database overhead.

Performance is all about structuring the task as close to the machine as possible.

You could add a row for each count. You could add more associated information with the count too, like date, why, etc.. Then that insert would trigger a debounced method that would query the event stack to give you a count. You can make periodic snapshots of the total to avoid O(n) table scans. This allows scale without row locking and a much more accurate accounting of when the counts happen. You rewind/fast forward, get a time series, whatever with the event stack. This is all part of the CQRS pattern.

https://martinfowler.com/bliki/CQRS.html

Outside of the DB: some kind of queue.

Could be a SQS or an in memory queue in your app (hand waves about mutiple nodes but that can be dealt with or use redis). Have a single processor that aggregates and does writes. You can use DLQ if something goes wrong to avoid data loss.

Inside the DB maybe just append rows for each add to a table without indexes. That way the engine probably just needs to write out a journal record and append but doesn't spend time on updating indexes. Then periodically read all from the table and aggregate into another. You could cycle between 2 tables.

These is eventually consistent in aggregations but transactional in source of truth. I.e. if you write you can be sure the data is stored, but may take time to aggregate.

Or Use a more exotic DB. That is distributed but still supports transactions.

Usually for this kind of use case you want both INCREMENT and DECREMENT. Using a random number for the "Slot" (AKA "Shard") seems like an anti-pattern. If the Slot were derived from a hash of the data itself, then DECREMENT would be much more straightforward.