The Republic of Texas, in terms of land that actually lay under its control, was in fact smaller than the current state of Texas, not larger. Its claims were certainly larger, but they were just that - claims. In reality the authority of the Republic didn’t extend much farther west than Austin or San Antonio, and it exerted little to no influence south of the Nueces River. To be certain, there were attempts to exercise control over their claims, but most were utter failures. The Republic of Texas was already in an economically and militarily poor shape and much of the leadership, including the first elected president, Sam Houston, recognized this. Houston’s end goal was annexation to the United States - the Republic stood little chance of surviving on its own. His successor, Mirabeau Lamar, had more grandiose plans, leading to an expedition to Santa Fe, New Mexico (a city that fell under the Republic’s broader claims) that ended up as a bit of a disaster.

To the south, the exact boundary line between Texas and Mexico was a large part of what sparked the Mexican-American War, because Mexico held effective control of everything south of the Nueces River (in effect, South Texas north to modern-day Corpus Christi). When the United States crossed that boundary Mexico saw it as a violation of the sovereign territory - which, to be honest, it kinda was. Abraham Lincoln famously argued against then-president Polk’s justification for the war, that “American blood had been spilled on American soil”, because the land between the Rio Grande and the Nueces was far from clearly American soil.

To the west of the zone where Texas exercised actual control was a region where neither it nor Mexico had any real power. The Comanche, a powerful nomadic indigenous people, held real sway here - a territory which includes most of West Texas, including the modern cities of Lubbock, Amarillo, or Wichita Falls, and they were able to defend their territory through any means necessary. In contrast to the more docile tribes of eastern Texas, who stood little chance against the violence inflicted on them by white and Tejano settlers, the Comanche were able to hold their own against the onslaught of white settlement. In fact, Sam Houston drafted a treaty which would recognize an official boundary between Texas and Comancheria, intending to cease the constant warfare of the Texas frontier. While this clause was deleted from the version ratified by this Texas Senate, it illustrates that Texan claims in west Texas essentially only existed on paper. The land was indisputably Comanche.

Once the Mexican-American War had ended, of course, the United States held effective control over South Texas and the parts of New Mexico claimed by Texas (notably, Comancheria still lay outside of American control). Texas “lost” its extra territory to the north and west as a result of the Compromise of 1850. It was not an uncontroversial land dispute. Slavery, in particular, was a major point of debate, as Texas was a slave state and many in the north opposed expanding African chattel slavery into the west (New Mexico already had an institutionalized system of indigenous slavery, but that’s not particularly relevant here). Texas eventually ceded most of its western lands, including what would later become the Oklahoma Panhandle, in exchange for a large sum of money. Of course, the state had never actually controlled said territories, but that part was irrelevant. For what it’s worth, had Texas kept the territories, it wouldn’t gain much in the way of population. The most significant major city would probably be Albuquerque, and some smaller cities like Santa Fe and Las Cruces. The claim extending into Colorado doesn’t appear to hit any noteworthy cities, as it goes west of the Denver-Colorado Springs-Pueblo corridor but stays east of Grand Junction. It might give Texas a few more ski resorts.