In terms of the classical sources, your observation is correct. The gemara does not discuss the serious cleaning we do in advance, and it does instruct that the bedika is performed on the night of the fourteenth.
However, the phenomenon you describe existed already in the time of the Rishonim. The Terumat HaDeshen[1] reports that many people would sweep the house a few days before Pesach, put a few pieces of bread in some rooms, and stop the bedika when they found them. He rejects this practice based on the Mordechai,[2] who maintains that sweeping the house beforehand is not sufficient. The Shulchan Aruch[3] concurs, saying that cleaning the house thoroughly before the night of the bedika, even with the intention of that serving as the bedika, does not exempt one from bedikat chametz on the night of the fourteenth. However, a few centuries later, the Sha’arei Teshuva[4] cites and justifies the practice you refer to. Does the practice conflict with the Terumat HaDeshen’s ruling, or has the situation changed so that the Terumat HaDeshen would agree?
The main reason the Mordechai gives for requiring a bedika even in a cleaned house is to avoid distinguishing between one bedika and another. In other words, Chazal instituted the time for doing a bedika as the eve of the fourteenth, and one should not say that his particular situation is an exception to the rule because he previously obviated the need. The Terumat HaDeshen already points out that if this were the only issue, it would have been sufficient to nominally fulfill the mitzva of bedika with a minimal bedika on the fourteenth. However, both the Mordechai and the Termat HaDeshen stress that sweeping the house does not do a complete job, as chametz may still remain in the holes and cracks, and it is therefore not a valid replacement for bedikat chametz. The Sha’arei Teshuva addresses this concern, justifying the actions of those who do a cursory job on the fourteenth by arguing that they clean very carefully beforehand.
The Magen Avraham[5] raises another issue. Chazal required that the bedika be done by candlelight at night or in an area with direct sunlight during the day. Few are careful to check all areas they clean in this manner. Furthermore, the Taz,[6] Bach,[7] and others say that even if one did a halachic level bedika on the thirteenth at night, he must repeat it on the fourteenth, the time Chazal instituted. The Chok Ya’akov[8] and Pri Chadash[9] say that one can fulfill the obligation of bedika on a night prior to the fourteenth, but the Ba’er Heitev[10] notes that this opinion is on the condition that one is careful that chametz be kept away from the checked areas after that point.
We can still justify the prevalent practice even according to those who say that the bedika must be done on the fourteenth. Only those areas into which chametz is sometimes brought need to be searched.[11] One can claim that areas that were cleaned and into which people were subsequently careful not to bring chametz are thereafter categorized as places that do not contain chametz. Thus, a superficial perusal of the house, during which one concentrates on confirming the assumption that indeed no chametz got into various rooms, may be considered as checking the relevant parts of the entire house.[12]
According to some poskim, the situation in which there is not really anything to check for on the night of bedikat chametz mandates the minhag to put out some pieces of bread in order to justify making a beracha on the search.[13] Others accept the assumption that one is checking to ensure that chametz did not somehow enter the cleaned rooms, and they justify the beracha on other grounds.[14]
It would be wrong to imply that all poskim fully accept the practice you mention. The Mishna Berura[15] seems to neither embrace it nor reject it. Rav Ovadia Yosef[16] says that it is proper to do a serious bedika on the fourteenth. However, our orientation is to instruct people to follow a prevalent minhag when it is justifiable, as this one is. (In contemporary times, many people have large homes containing so many possessions that it is barely feasible to do a proper bedika in one night, in any case.) If one wants to be more stringent, that is his prerogative.