• WARNING: Tube/Valve amplifiers use potentially LETHAL HIGH VOLTAGES.
    Building, troubleshooting and testing of these amplifiers should only be
    performed by someone who is thoroughly familiar with
    the safety precautions around high voltages.

Printed circuit board designing?

Yes, absolutely. Many new amps are built that way.

For the valves, rather than use a header connected by a cable to the valve socket on the chassis, ( which is what I think you are describing ), you can use a pcb type tube socket instead. Or what I have done is use a regular old chassis type tube socket on the pcb with short jumper wires connected to points on the board.
Yup - I've personally chosen a PCB WAY bigger than I need, so that'll allow me to have a bank of connectors at one end (power), another at the other (valves), and keep all the components over by the power end, and keep the tracks well away from each other - ie. all the components are about 2x2", board is 9x12"....
Since you said component clearance was not enough with chassis top. It was suggested to mount the components on the opposite side of the board. Meaning mount the components on the bottom of the PCB and mount the sockets on the top. Clearence issue will be solved.

See how the sockets are mounted on the opposite side

It allows you to panel mount the tubes when using a PCB.
That's actually not my amplifier, I just used the picture to show what I was talking about regarding mounting the capacitors on the opposite side of the board from the tubes.

Usually if using DC heaters, I'll run the lines on be board, because the rectifiers, filter cap and regulator (if using one) is mounted on the board. Even though it's DC, it's best to avoid running the lines too close while parallel to signal lines. And also, if it's a double sided board run the heater V+ and it's ground return overlapped on opposite sides of the board.

If using AC heaters, I twist the transformer heater wires and solder them strait to the socket pins.