To an extent, this is also handwaved in the light novels as well - the specifics of how the siblings were/are protected are not disclosed. However, there are many references to the Yotsuba family's incredible power and influence, which allows us to infer - to a certain level - what's going on.

While the Japanese public may well have been interested in the heirs of the "10 families", the Yotsuba are a special case. More so than the other clans, (such as the Ichijou, Saegusa and Juumonji) the Yotsuba are an inherently secretive family, even before considering Tatsuya and Miyuki's special case:

"Because the Yotsuba Family, who maintain a systematic policy of secrecy, would never invite a large number of outsiders as guests... the siblings had brought themselves to the mountain village that was not recorded on any map..." Volume 8 - Reminiscence, Chapter 1

And

[...] not even Miyuki knew the number for the direct line. The information control surrounding the Yotsuba Family was several times more intense than the government, and that was no idle exaggeration by those in the know. Volume 9 - Visitor I, Chapter 6

Prior to the sibling's enrollment at First High, they too would have naturally been caught inside the familial policy of secrecy, meaning their chances of being discovered by outsiders was slim. Even during this time, Tatsuya is under strict instruction from his mother to remain inconspicuous:

"Tatsuya, it’s as you heard. You’ll be accompanied by Miyuki san on your tour of the base." "Yes." "Just keep one thing in mind. In public, don’t go using honorifics or the like when addressing Miyuki-san. Rather than ‘Ojou-sama’, just ‘Miyuki’ is fine. Actions which could lead to the discovery that Miyuki-san is a Yotsuba Head successor are prohibited." Volume 8 - Reminiscence, Chapter 9

In addition, the military assists the Yotsuba in keeping Tatsuya's existence under wraps, as his power as a strategic-class magician is important for them to guard as well:

"It is most likely some of them will link this with the annihilation of the fleet the Great Asian Alliance dispatched three years ago. It would be a very undesirable outcome for us to have Tatsuya-san's identity to come out." "I fully agree." Seeing Kazama's nod, Maya gave a broad smile indistinguishable between real and fake... Maya's negotiation with Kazama was steered to obtain the most satisfactory result for the Yotsuba. Volume 8 - Reminiscence, Chapter 15

Working in collaboration with the military by extension means working with the "Electron Sorceress" Fujibayashi Kyouko , who can manipulate electronic information to an unfathomable degree, creating an almost watertight seal around the siblings information.

This leaves a situation where even Kokonoe Yakumo, a renowned information broker, cannot uncover any of their background information:

Tatsuya squinted his eyes. "Our situation as well?" Yakumo laughed heartily. "I've inquired into it, but wasn't able to unearth any noteworthy secrets. The intelligence manipulation surrounding the two of you is truly invincible. Or should I say, worthy of its high reputation." Volume 2 - Enrollment II, Chapter 9

The same is true for the information network available to the Kitayama family:

In response, Tatsuya received it with a magnanimous look. “Who the devil are you? With Kitayama’s…… The information network from the “Corporate Alliance” couldn’t even find your Personal Data!” Volume 12 - Double Seven, Chapter 2

When it comes down to it though, the people that have the power, influence or luck to uncover the truth about the two would be too scared to both continue their investigation or disclose the information to anyone else after hearing the name 'Yotsuba':

The Yotsuba name had even more impact than the Saegusa name. The political utility of the Saegusa name was higher, but the Yotsuba name incited much more fear in people with influence. Volume 12 - Double Seven, Chapter 6

This fear is also shown when Erika stumbles across Tatsuya's secret accidentally, and warns her brother against investigating him too much:

“I see…… I understand. Then let’s proceed along these lines.” ……Fortunately, she was able to convince her brother without dropping the name “Yotsuba”. Volume 11 - Visitor III, Chapter 15

Even those not in the higher echelons of magicians should be rightfully wary of the Yotsuba family - after all, this is the clan that previously took down an entire nation in under 2 years! Furthermore, at present, they're quoted as being even stronger than then:

[...it] was utterly incredible that they could have forgotten such a harsh reminder in the last 20 to 30 years. Not to say the might of the Yotsuba Family – “might” that hinted at influence but was more along the lines of violence – was far greater now than it was in the past. Volume 11 - Visitor III, Chapter 15

For the points you raised:

This isn't covered in the light novels, so I can't reference anything in particular, but given the other information, it's safe to assume that anything the Yotsuba wanted to keep secret (like children from a marriage) could be kept secret. Their existence isn't erased, but all the data relating to them has been heavily modified, to the point where only the bare essentials are available, and there's nothing that could link them to the Yotsuba:

"After all, I can’t attend high school without Personal Data.” Tatsuya’s answer made perfect sense. However, Benio felt that this was the rub. “Please don’t underestimate adults. It is true that your PD has the lowest level of information, interlaced with extra information that complicated the picture and a few negative reviews, so it wasn't overly perfect." Volume 12 - Double Seven, Chapter 2

When involved with FLT, Tatsuya goes by the alias 'Silver', and not his family name 'Shiba', which would help against identification from most staff or customers. In addition, while FLT is "owned" by Tatsuya's father, it's actually under Yotsuba jurisdiction, meaning they would have a careful eye on the situation, especially Tatsuya's colleagues who would be more likely to accidentally uncover sensitive information.

Like FLT, this hotel was one of the dummy corporations that was controlled by the Yotsuba Family in the background. Volume 10 - Visitor II, Chapter 9