The center tune meter stops being a center tune meter when you're on a preset. It displays the tuned frequency instead, look at the markings at the bottom of the scale. There actually is no way to tell from the meter whether the preset is pulling you off center tune, though you can tell by monitoring from one of the rear panel jacks or when connected to it with test gear. You CAN tell with servo tune though, try turning the Servo switch off, tune manually to a station and center the meter, then switch on Servo. If it pulls the meter off-center, one of two things is happening: either the servo adjustment is off, or the alignment points that dictate the meter indication are off and you weren't really center-tuned. You may be able to tell which by checking to see whether max signal on the lefthand meter also coincides consistently on different stations with indicated center tune on the righthand meter - not that it always is but more often than not, it will be if the alignment is correct. If you find that max signal is obtained with the center tune meter off to one side or the other, or that you hear the signal drop out/go to static quicker in one direction than the other, it's probably out of alignment.
The orange caps are low-leakage type and they're original. All of it looks original, but that battery might not be - they can leak that badly in 10 years or less, so it could have been replaced before.
On the solder joints: you haven't got it apart far enough to see the ones that cause the most trouble for the tuning circuit. Those will be on the two tuning system boards that are stacked with the solder sides facing inwards, which you will need to remove and separate in order to replace the battery and the presets. Take the whole stack out (which means also pulling the front panel forward enough to free up the preset board), then carefully separate the two. All of the header joints need examination, same for the connector to the preset board (as well as the other side that's on the preset board). Also make sure the header pins themselves are clean.
Aside from that, the problem areas for solder otherwise are the main board at the power supply, at the headers that connect the selector RF box off to the right, along the front panel switches & pots where they solder to the main board, the header points for the lighting board (there's a circuit jumper that runs across that), at the muting relay, and the headers where the plugs from the front panel plug in. Bad connections in the shielded boxes are rare.
Plan on replacing at least these five caps: the three largest in the power supply (1000uF/63V, 1000uF/50V, and 220uF/63V but I use a 220/100 for that one - there are good Nichicon caps with matching lead spacing, Mouser has all three). That is important for keeping the regulated and unregulated rails on the power supply quiet/stable. Also the two larger caps on the stereo decoder board (1000uF/35V low profile - there's a Nichicon series that fits well, think I have used UHE for that), and make sure those two haven't leaked into the PCB under the caps. If the top of the PCB is blackened after you remove the caps, and/or there is erosion on the solder side, clean all that up and in bad cases you will need to cut little trenches into the board top surfaces between the solder pads to break any carbon leakage across the caps (the board surface can become conductive if enough electrolyte has oozed into it).
Overall it looks like a typical unmolested middle production 3001A; you can tell it's not early because on the earlier ones that twisted multiharness that connects the main board to one of the tuning system boards would be a hard header board (like the lighting board); they went to this harness design later, which is more reliable - less problems with solder joints by getting rid of the hard-mounted parts & header connections.
John