Why are there lots of empty tiles in the city centre?

You appear to have 'Hide buildings' turned on. Some buildings (parks, fountains, etc.) have no foundations at all. Some of these do cause pavement to appear, but not all. Have you queried the empty tiles to make sure they're actually empty (and not just parks)? Iff so, proceed:

In order to understand what's happening, you need a more detailed understanding of the game's housing placement algorithm (see fn. GrowTownInTile).

The game will execute a 'walking' function that will follow the roads of the town by randomly picking an adjacent tile with each iteration. It starts at the town's seed tile, directly located under the town name. If the tile picked is already occupied by a building or road, it will go explore along the road.

The function will then, at each intersection, make a random choice where to go, with each single tile walked it will try out an additional random tile next to the current road tile to build a house. It will not back out on itself. Since version 1.9 it will also check diagonal tiles (build houses on corners).

Some of those empty spots you note aren't adjacent to a road. So the house placement algorithm will never find them, and they will remain empty. You may sometimes see spaces not adjacent to a road built up by a town. this is usually the case due to 2x1 or 2x2 buildings taking up such a tile, but the game won't explicitly select such bulidings to fill the tiles, it merely happens due to coincidence.

If it walks around for a long time without finding a location (the total time it attempts to search depends on the size of the town, longer towns will explore for longer) or it runs into a dead end, it will stop (fail) and execute again in 70 ticks. As this is the maximum growth speed, it stands to reason that occasional failures to find a spot to build have a much larger effect on the growth speed of bigger towns, as for a big town a 50% failure rate is a 50% reduction in growth speed, while for a town that grows once every say 30 days a 50% failure rate results in only about a 3% reduction in growth speed.

Try experimenting here with the game's town growth algorithm. Here's a result of one of my experiments. I ran a town on it for a while (it's a really accurate model of openttd's algorithm, pre-1.9)

Notice how the random nature of the movement caused the algorithm to loop in on itself and not find a spot to place a house, even though close-to-center of town spots next to a road exist.

A town with a lot of intersections thus will eventually get dead space next to roads that takes a long time to fill in, simply because the random routes the placement algorithm takes just happen to miss these empty spaces. Here's another (more extreme) example:

In 320 attempts, this town managed to build 18 buildings. Without all the intersections and dead ends, it would grow orders of magnitude faster.

Create less intersections and loops in your road system to prevent this. The optimum town growth is reached by building in a forking tree or spiral pattern, with no loops (ways to move in a circle) at all. Since

_grow_town_result = 10 + t->cache.num_houses * 1 / 9;

(note, this is for 2x2 and 3x3 grid layouts, other layouts may search for far longer). our spiral needs to have at least 9 available building tiles for each tile from the center. As each spiral arm gains about 2 tiles per road tile, the spiral needs at least 5 arms for the town to keep growing.

Also see OpenTTD City size declining and my answer to Is there any benefit in funding new buildings while the town is still funded?.