Yes, it’s technically possible, but traditional DSLR sensors had tradeoffs that made full electronic shutters less attractive.

With many CCD-style electronic shutter designs, each pixel needs extra storage/shielded circuitry to hold charge when exposure ends. That extra circuitry takes up space that would otherwise be light-sensitive area, reducing fill factor. The result is typically lower sensitivity, more noise, worse low-light/high-ISO performance, and reduced image quality.

Older compact cameras often used this kind of sensor-based shutter because it was cheaper and acceptable for small-sensor cameras. But higher-end DSLRs prioritized image quality, so mechanical shutters remained preferable.

CCD sensors could be used more easily for electronic shuttering, but they also had issues such as blooming. CMOS sensors, which became dominant in DSLRs because of their other advantages, were historically more difficult to start/stop electronically in a way that worked well for the whole sensor.

As for “record everything for a second and choose later,” that would require storing huge amounts of per-pixel data continuously and doesn’t avoid the basic sensor/readout limits. It’s not a practical substitute for a true shutter.

So the short answer is: electronic shutters are possible, but older DSLR technology involved image-quality and sensor-design compromises that made mechanical shutters the better choice.