Did you replace the focuser on your 628? If so, did you use the existing screw holes and was the new curvature ok?
Secondary mirror - for the 628, is there a 1.3" to 1.52" to 1.8" advantage to be gained? I am no maths expert when it comes to obstruction versus light gathering!
Greasing - Lithium is ok for everything...worm, plastic shaft bearings, circles ?
I did not replace anything, especially the focusers, on any of my 826, 628, or Starfinder GEM 6" and 8" scopes. I have no need to modify major parts of a vintage scope I'm fixing up because I have other scopes to use. The more major an aspect is, the less likely I am to change it. I do a lot of mods when I fix up a scope, but most are small things like hardware or clearances. I will make major changes on things that never worked from the factory or were just horribly done. I keep mods minimal on major flaws, especially infamous flaws, to preserve the factory flub as much as possible and be user friendly. This all becomes a subjective shade of gray, to each his own. A tall focuser is OK with me if it works OK and was factory, a secondary that vignettes on axis rays is NOT ok with me (major engineering flub).
You can stay with your original 1.3" secondary if there are no other issues with it besides size. I would go with a 1.5" if I was going to change it for other reasons. I measured my 628 (yours might have a different height focuser). I have 7.25" from the secondary to a point 1/4" above the racked in focuser. The math shows that a minimal size secondary capable of using the whole 6" primary only needs to be 0.91" minor axis. Another way to state this is that the secondary only needs to be .91" for on axis rays. On the practical side, you need to be bigger if you view anything besides stars or planets. The original 1.3" secondary will let you see the full moon with no light loss at the edge. ONLY for you techies: A 1.3" secondary gives a 0.56" fully illuminated field. A 1.5" secondary gives a 0.84" fully illuminated field.
I suggest being leery of grease request. You are in a forum of some upper end folks that will spout specs and fancy expensive grease and lubes and such along with how awesome their product suggestion is. I use garage sale finds. You have a 6" telescope, not a factory custom 1928 Pierce Arrow automobile. Modern lithium grease should work for a generation or two, it has in the past. You can do better than lithium now. Get a small tube, you don't need a can unless you do your own car work. I find a good axle grease to work around the house in general. I use butter on door latches and strike plates because the kitchen is centrally located.