Does the type of strainer built in a teapot matter?

I have used several types of integrated strainers and in my experience nothing is better than a wire mesh. The holes and slits in glass and plastic strainers are too small and too few to allow good water circulation, so you have to move the strainer around to get the tea to brew. Cleaning-wise, slits are the worst as their tapered ends trap leaved which can be annoying to remove. The plastic strainers are probably the easiest to clean, with wire being in the middle.

I'd personally go with the wire mesh type as it brews the best tea.

For some teas, the strainer does matter. For a particular variety of green tea that I was considering importing years ago, the producer recommended a certain type of fine mesh strainer for best results, as apparently the leaves were just the right shape to clog up the "standard" Kyuusu-style mesh strainer. (Kyuusu is a type of Japanese teapot with the handle on the side instead of opposite the spout).

With black tea, I found that those glass teapot strainers with the little slits in them worked fine, but were a pain to clean. For Japanese green teas, though, they were a disaster, as tea leaves got stuck in them and the infusion never got very complete, and the water refused to drain in a reasonable amount of time. As a result, I refuse to buy another glass teapot with those little slits, because they were too unreliable for anything but large leaf black teas or certain "fruit teas" (as they were called in Germany).

Korean style one-person tea cups with large holes in the strainer component work reasonably well for green tea (and are designed for green tea), but often require toothpicks or bamboo skewers to clean if tea gets stuck in the holes.

My favorite teapots mostly use the Japanese-style steel mesh, though I have had good luck for certain teas with built-in clay strainer holes if they brew small quantities of tea.

Yes, I think that they strainer does make a difference. I would say that they are best for nearly any tea leaf is the fine wire mesh strainer. Plastic mesh is probably second best and the glass or plastic with a few holes as the least desirable.

When brewing whole leaf tea, you want to make sure that the leaves have enough room to bloom and open. If the strainer or teapot is not big enough for your tea leaves to open you will not get the most out of your tea leaves.

The fine metal mesh is probably the best, as it is the most robust and can cleaned easily. The mesh is also quite fine or comes in different degrees of fineness. The metal is easily bleached or otherwise cleaned. The plastic will become brittle and it a bit harder to clean over time, but is probably the cheapest to replace. The solid glass or plastic with slits will be easy to clean, but usually does not have enough space for the leaves to bloom, and will be the most fragile.

I would suggest that if you have unglazed teapots, that they be used with only one type of tea. An example is one teapot for just Chinese jasmine, one more for oolong, another for pu-er, and so on. And this is especially true for more expensive teas to be used in the same teapot. The oils will be infused into the teapot and strainer. And no mixing of flavours from different teas.

So going back to the OP, I would say that a big, fine meshed, metal strainer is best. Big, so that your tea leaves can open up. Fine meshed to strain all the leaves out of the water/liquid. Metal as it holds up best and can be cleaned easily.