No reason you couldn't do both, and just have the smartphone app interface with the same system the in-house tablets do. (And not everyone will leave a smartphone.)
Menu systems are simple, and don't need much in the way of processing power. They don't need to be full-featured web browsers, or be able to run high-end games. An e-Ink screen would be enough for most places. This approach even makes sense from a recycling point of view - just use screens from old iPads or Kindles for the in-house tablets.
The value in high-end processing is the multimedia upsell. Replacing a paper menu doesn't require much processing power, but it's worth a lot to increase the check average and/or improve margins by promoting higher margin items.
Menu systems are simple, and don't need much in the way of processing power. They don't need to be full-featured web browsers, or be able to run high-end games. An e-Ink screen would be enough for most places. This approach even makes sense from a recycling point of view - just use screens from old iPads or Kindles for the in-house tablets.