Why does my old Kenwood only come alive after it warms up?

That warm-up-then-works pattern is classic heat-sensitive failure, and it ussually points at either a dried-out cap or a cracked solder joint that only makes contact once the board expands. Before you replace anything, try localizing it with heat and cold. Next cold start, hit sections of the board with a hair dryer on low for a few seconds at a time. If the radio wakes up early, you've found your neighborhood. Or do the reverse once it's running: blow cool air on one board at a time til the fault comes back. Costs nothing and it beats guessing. First thing I'd actually measure is the power supply output. The service manual for the TS-430S is freely available online and it lists the test voltages right on the schematic. Check the main rails with your multimeter cold and again after it warms up. If a rail comes up low or slow at startup, the filter caps in the supply are the usual suspects, and those are the ones I'd replace first regardless.

On shotgunning all the caps: I'm in the cautious camp on that one. Pulling every electrolytic on an old rig with a cheap iron is how pads lift and traces get damaged. What I'd do is replace the supply section and the display/control board caps, since your symptoms live there, and leave the RF boards alone unless they misbehave. A cheap ESR meter, like thirty bucks, lets you test caps in circuit and find the marginal ones without a scope.

While you're in there, find the memory backup cell on the control board. Some of those old lithium cells leaked and ate traces underneath. If there's green crust around it, clean it well before you solder in a fresh one.

And one tip from experiance: go slow desoldering old caps. Heat each leg, work the cap loose gently, and add fresh solder to help transfer heat. Rushing that step is what kills boards.

Just went through almost the same thing with an old receiver, and it turned out to be a cracked solder joint on one of the display board pins, not a cap at all. Reheated the joint, added a touch of fresh solder, and the wierd cold-start behavior vanished completly. So don't get tunnel vision on capacitors, the heat/cold trick the other guy described will save you from swapping good parts. Also take a phone photo of the board before you start desoldering anything, saved me when I forgot where a wire went.

I'd add a photo step before touching anything. Take clear pics of the board top and bottom, both sides of every cap, so you know original orientation and what the pads should look like. Saved me twice when I lost track of a wire mid-job. Also on the display fault, check the ribbon cable and connector seating while you're in there. On mine the intermittent was just a grimy edge connector, wiped it with isopropyl and a coffee filter and the flaky startup went away. Cost me nothing but ten minutes.

Watching this with interest since I'm in a similar spot, just with an old stereo amp instead of a radio. Same deal, sounds awful for the first while then fine once it's been on. I found that tapping the board gently with a screwdriver handle while it's acting up can help pin things down too, if the sound cuts in and out you're close to the culprit. Don't know if that works the same on a radio but worth a try while you're doing the hair dryer thing. Where do people usually find the service manual, is it just a pdf floating around or do you buy it?