Almost all gutter covers don't protect your home against ice formation. Heat wire and heat tape, besides being inefficient, are very dangerous. You will see icicles sooner with all gutter protection.
Instead of the ice building up in the gutter and eventually overflowing, it will be sitting on the gutter cover where it will form icicles.
Gutter protectors that cover the first row of shingles and attach under the second row of shingles will provide some protection against ice damage on the roof. This is because ice damage to the roof occurs when ice gets under the first row of shingles which is the roof's first line of defense. These systems block the ice from getting under the first row of shingles.
Gutter guards that mount on the fascia or sit in the gutter (such as screens or panels with vertical holes) are more likely to cause ice damage on the roof. These types of gutter covers are mounted underneath the first row of shingles. Snow sits on top of them and ice water forms layers over the snow. Eventually the ice gets a boost up and pries up the first row of shingles destroying the shingles and causing leaks in the roof and ceiling.
In the northern climates, 3x4 oversized down spouts are a great value to enhance any gutter or gutter cap system. During the warmer months, overflowing in the gutters is minimized as you are doubling the capacity of water to flow through the down spout. During the colder months, it takes longer, if at all, for ice to block a 3x4 down spout. Whereas a 2x3 down spout will quickly clog with ice.
Heat tape and heat wires can be bought at the hardware store. They are usually installed by the homeowner. There is a good reason why we have licensed electricians. Electricity is dangerous. Many heat tape and wires are not temperature regulated and can potentially catch structures on fire. After all, the asphalt in the shingles is an oil derivative and is flammable, along with any wood structures that come in contact with the wire. The insulation covering these products is thin and easily nicked which increases the potential for fires. These products are usually hooked up to a plug in to an outside outlet. It isn't a good idea to have situations to where wires constantly come in contact with water. These systems are accidents waiting to happen.
The "plug in" component causes other issues as well. Since they are not connected to their own breaker, they limit power for electric appliances, and they dim Christmas and other holiday decorations on that circuit.
The biggest issue is that heat tapes and wires are just not efficient at melting ice. Only the area immediately around the wire gets melted. Ice and snow quickly overwhelm these systems and you will see tunneling through the ice mass. And the ice issue is still there.