If by old school you mean tried tested, proven, reliable and adaptable then I agree.
I have never taken one of these 'distribution' amplifiers apart (straight in the bin - no messing) but I assume they are an amplifier combined with a splitter internally so little different to what I suggested but with disadvantages. Your post implies that more signal / amplification is required when using a splitter which is not the case.
Scenario one. A site with 250 ish tv points. 21 12 way 'distribution' amps? No 1 launch amp (this is an actual job I did where I used two launch amps for practical purposes) plus taps and splitters.
Scenario two. Four way 6dB 'distribution' amp in situ. Customer wants two extra points. Split a leg reducing the signal level by 4dB on the split legs? Chuck the 4 way amp and replace with a 6 way one? Or replace a four way splitter for a six way one at a cost of a couple of pounds and only have an extra 2ish dB loss on the legs and, being passive, reuse the splitter).
Scenario three (as this is an electricians forum). Four way amp left connected by the electrician, aerial fitted but pixellation on tv's, carrier to noise ratio too low so we need to amplify correctly (ie as soon as we can, 1m from the aerial with a masthead amp) unfortunately there is a distribution amp in the way not allowing power pass. Guess what, chuck it in the bin, replace it with a splitter and take advantage of the power pass leg(s).
I am not saying that you cannot use one of these things, I question why people use them as all I see are disadvantages and no advantages. An electrician could carry 1 variable gain amp (£10), a two way, a three way, a four way, a 6 way and an 8 way splitter costing £15 ish for them all making a total cost of £25. Include a masthead and psu and you are still sub £40. This will see you through most situations. Carrying a two, three, four, six and an eight way 'distribution' amp (probably two of each as you would require different gain levels) would be well over £200 and just would not be as versatile, plus your van would be full of totally unnecessary items