Back to removal of the remnants from the caulk, have you considered a 2 part acid teak cleaner? I'm not sure if it will remove the varnish bits, but you many want a good cleaning of the teak before you reseal anyway. Maybe worth a shot.
Suggestion for trial #1
Water is the ultimate destroyer of any
topside coating, even Awlgrip and LPU urethanes.
• cover a small area with a thick layer of paper toweling
• cover the toweling with plastic sheeting and SEAL the edges of the plastic with tape, etc.
• open a small area of the 'sealing' and add WATER plus a few drops of dishwashing detergent, re-seal.
• allow to SOAK for several days/weeks, re-wet as needed. Soak time is important.
• remove the covering, etc. and scrub the caulk lines.
Suggestion #2 for a trial in a small area:
A. obtain trisodiumphosphate-TSP crystals .... from a paint hardware store BUT be sure that is REAL
TSP and not 'substitute' TSP.
B. Mix up the TSP with hot
water and so that the mixture has EXCESS crystals in the bottom of the 'bucket'.
C. Apply the TSP solution to the surface and immediately cover with a sheet of plastic film. Allow to soak for several hours under the sheet. You may have to intermittently re-wet the TSP/teak.
D. Scrub (only) the varnished caulk lines with a moderately soft brush, dont scrub the teak.
The TSP is highly caustic and most times will loosen / break the adhesion of paint/varnish on wood if there is sufficient 'soak time'. However, the TSP will also extract the tannins in the UV affected surface layer of the teak ... and produce a diarrhea-like brown 'slurry' as it dissolves the 'dead' sun destroyed teak surface layer; but, will leave the 'unaffected' teak undissolved.
To restore the natural 'hue' of the teak once extracted with TSP: rinse several times to remove the 'goo' (dissolved 'dead' teak), rinse again, let the surface thoroughly dry out, then 'bleach' the teak with oxalic acid (paint/hardware stuff - crystals in pint containers ... mix as you did with TSP)
Obviously, TSP and Oxalic are 'chemicals' so you may not be able to obtain such in states that have totalitarian environmental statutes vs. 'consumer usage' of even simple 'chemicals'. You can get these online if you absolutely need to.
• TSP should NEVER be mixed with oxalic acid .... you MUST thoroughly rinse between each chemical application.
• Wear protection: goggles, gloves, plastic apron, etc. Oxalic acid quickly absorbs through your skin and reforms as sharp crystals in your kidneys ... you MUST wear 'protection' to keep it off your skin.