Batumi can work very well for remote workers, but it depends on expectations. If you want a city that feels visually interesting, relatively relaxed and easy to navigate, it has a lot going for it.
At the same time, remote work life depends on more than aesthetics. Reliable routines, comfortable cafés, apartment quality, street noise and your own work style matter far more than social media impressions.
The good part is that Batumi is compact enough that you can build a rhythm without wasting huge amounts of time commuting. Once you identify your core places, daily life becomes efficient quite quickly.
The key is not to search for a perfect digital nomad fantasy. It is to build a setup that fits your actual habits.