Regular lead solder with rosin flux should work fine here. The board is probably made with lead-free solder to comply with regulations but for a repair lead solder is easier to work with.Thanks for the tips and feedback jtr. What brand/type of solder should I use for this kind of repair with respect to tin/lead and rosin/flux, etc? I'm going to dig out the soldering iron I have to see what kind of wattage it has and whatever kind of solder I might have with it. I had this Stahl Tools SSVT Variable Temperature Soldering Station on my bookmark list for some Arduino projects I was looking to get into but hadn't yet. Its max wattage is at the minimum you recommend. I could pick up one of these desoldering pumps to clean out the old solder or some of the soldering braid like you mentioned. This video looks pretty straight forward on cleaning out the old solder with the braid.
Are the replacement capacitors I linked to look ok in terms of specs? You mentioned the polymer ones but I haven't been able to find an appropriate one to replace these.
The replacement caps seem OK to me. It took 7 years for the old ones to fail. Assuming this is the problem, and your repair fixes it, the TV will most likely last until the TVs you're interested in come down in price. I looked into 2200uF, 10V solid caps but they don't exist at Mouser. The highest voltage for 2200 uF solid caps is 6.3V. They do have 1500 uF, 10V solid caps but without knowing more about the circuit I couldn't recommend using a different value. Also, the ones they have are surface mount, not the type with leads, so they wouldn't work here.