Go here: http://www.antennaweb.org/Address.aspx and put in the zip code. It will show a map of the TV stations nearby and the RF channel they are on. The FCC tried to free up the low VHF and high UHF TV channels during the digital transition, so the existing antenna may not cover the needed RF channels.
The 300 to 75 ohm balun should be installed at the antenna and a 75 ohm coax downlead used from the balun to the TV set. You can put the balun at the TV, but the old 300 ohm twinlead will not perform as well as new coax especially at UHF.
Actually,all things being equal,the twin line will be less lossy than coax--especially the stuff which is currently used.
Most of the "RG6" & the like,is really designed for cable use,where the desired signal is quite high level,& the extra screening is required to prevent radiation into other services.
It looks great,compared to the shoddy looking TV coax which was commonly used in earlier times,but its losses are higher at UHF ---about the same as "real RG59".(not the crud they try to sell as RG59 now)
The "shoddy cable" was air cored,with much better loss figures.
I had occasion some years back to try RG59 for a long run in a TV distribution system after we ran out of the cable we had been using---the signal was unuseable with RG59!
All that said,the stuff currently used is cheap,rugged,& in most cases the additional loss is of no consequence.
If the original installation is 40 years old, most of the connections will have corroded,the insulation crumbled off the twinlead,& any splitters will be blobs of corroded copper.
Digital TV is very rugged,so the stations you are getting may be from random pickup on a broken length of twinline.
When I bought a Digital TV,we got some stations OK on the old antenna & coax.
The old antenna was quite literally falling apart,so I fitted a new one & temporarily connected it to the old feeder.
NO change!--still the same group of stations.
When I fitted the new cable,I got all the stations OK.
I'm not sure how they did it in your country,but we stopped using the low VHF channels ,so a Digital antenna is quite compact compared to the old one.
If you are going to climb around on the roof,you may as well fit a new antenna & cable & be done with it!