I wouldn't shotgun all of the replacement caps unless they look old. You got a raster before, so your TV may be pretty close to working.
"Old" is a relative term, I know. If the non-electrolytic replacement caps look like these old plastic-coated paper caps (perhaps with different colors), then I would get rid of 'em:
But if they are like these contemporary orange drops and little yellow guys, I would leave 'em alone for now:
Other modern caps may be white tubes rather than yellow, or look like tan Chiclets.
I would not re-replace the electrolytics just for the heck of it. If your set is still producing some HV, and many other voltages are in the right ballpark, the electrolytics can't be that bad.
Bleeder chain: There is a bunch of high-value (1 megohm and higher) resistors between the picture tube and the vertical & horizontal circuits. Look at the lower right in your schematic. If you don't have a HV voltage tester, you can still test these with an ohmmeter. These values are not super critical, but who knows, one of them may have burned up and gone open. When I checked these in my Admiral 19A11, most were reasonable, but one of the 4.7 meg resistors had drifted to 9 meg, so out it went.
Phil Nelson