Is there a way to make a generic cheese sauce?

'Cheese sauce' is really anything semi-liquid that tastes a bit like cheese & can be poured or dipped, depending on how liquid.

Some fast ideas:-

Camembert, brie etc - put it in the oven for 20 mins. Cross-cut the top, dip.
tbh, you can do this with most cheeses, just the French-style crusted cheeses provide a cool looking container.

Generic roux - oil or butter in a saucepan, add the same quantity of flour. Combine 2 mins, add milk slowly, stir. Add cheese, stir, serve.

Welsh rarebit - throw cheese, flour, milk, beer, worcestershire sauce in a pan, heat & stir until it's homogenous.

There's not really much you can do to hot cheese to spoil it ;)
The only two cheeses I can think of that don't work are haloumi & paneer, because they're both pre-cooked & don't melt.

Your "cheesy bechamel" is a classic cheese sauce. It works best with a fairly full-flavoured cheese in the usual proportions, but you can adjust the ratios successfully for flavour and texture. Served hot it's good as a topping or dip, though keeping a dip warm can be a hassle.

With the pictured cheese I'd probably make the bechamel a little on the thin side, then add plenty of cheese, but if it gets too thick you can always add a bit more milk. Personally I'd also add black pepper and a pinch of mustard powder (not enough to make it taste of mustard, just to enhance the flavour). I don't infuse the milk when I make cheese sauce this way.