Does Wider Foil Give Greater Strength

The strongest part of a stained glass panel, whether leaded or copper foiled, is the glass.  The weaker points are the matrix that holds the panel together.

Of the matrix, the solder is the strong part.  The copper foil is weaker (and much thinner) and the adhesive is the weakest part of all.

A wider bead gives more apparent strength, but on the surface. It provides a broad line to grasp the glass.  But wide beads are often not what is visually desirable, nor practical.  And the wider the bead, the more solder will be used.

The most important part of a panel is the thin fin of solder between the top and bottom of the solder beads.  This is the connector between front and back. The strength of the whole panel depends on that fin.  So, it could be argued that very closely fitting foiled pieces lead to a weaker panel than loosely fitting ones.  I would not argue that, but it is important to have that connector of solder between the surface beads for a panel to be strong.

The solder connects the two solder beads together and forms the matrix which holds the panel together.

Here the fin of solder is a little thinner, so the matrix is marginally weaker

For larger panels, reinforcement will be required, either between the glass pieces, or on the surface.  The fact that reinforcement is so often used in the gap between the pieces, is confirmation that the fin of solder between the front and back is very important.