As someone who went through what you are going through, I would honestly recommend the xrgb-mini (framemeister). Don't let the price scare you off, the steep price is worth the investment as it only produces about 2ms of lag and looks beautiful on modern televisions.

This is the Rolls Royce of upscalers, just don't fall down the SCART rabbit hole.

A high quality capture card like the Elgato 60HD Pro + A decent upscaler (device that converts the A/V to HD) the best of which is the Framemeister but any decent one will do.