"Good tube Integrated Amps"

I’d add the Ayon Scorpio to your audition list.

PrimaLuna Dialogue Premium HP is right around that price range.  Read the reviews.

Line Magnetic Audio 218ia SET amp. Very nice sound. I have the 216ia and had the 218ia for a short time. It was very sweet sounding.

Hi there
for 5k you can get a used audio note meishu (the sweet spot in the audio note UK lineup). It is a very highly regarded integrated amp. I've had mine for 20 years and i get everything i want out if it in a 350 square foot room and 93db speakers. You will need fairly efficient speakers unless the room is small. 

pear audio blue Classic intergrated at 4K ,