No, there isn't a downside, and though I always book a flight I would intend to fly if all else fails, then this is an acceptable way of doing things. The catches are:

1) As you indicate, flights can fill up at the last moment, particularly if there are irrops or a flight gets cancelled: displaced passengers can instantly take all the available space for that day. Moreover if the flight you are really interested in is much more expensive, that will be the one to get filled.

2) Assume you can't change airport (between LCY/LHR/LGW). It's a bit more complex than that, it is sometimes enabled when there are irrops, also if the fare difference to the other airport falls below what you paid, but you won't know that until the day, so assume it can't happen.

3) Best to do any changes via BA.com rather than the App. The App can still be used but the clunky refresh process for this is annoying. Look for flights in My booked flights rather than via the main page.

4) If you are changing the return, as opposed to the first leg, you can assume the ticket will need to be revalidated. The automated process doesn't work. So don't work on the basis of change flight/check in/select seats. It may require a call to BAEC to do the revalidation, then OLCI can work on BA.com (or the App, subject to point 3). Or do it at the airport, it's easier.

5) The free day of change kicks in at precisely midnight at the departure city location - put like that since Plus and Flex fares are available to/from Ireland and Mainland Europe too.

6) Don't totally discount Basic (HBO) fares, yes there is a change fee of £60 plus change fee, but if the flights are filling up it may be a more calculated process to pay less and then pay more if the planets align. Moreover on Basic fares if you cancel the ticket, you may get a very high percentage of your ticket back. So on an £69 ticket you may get £43 of it back despite being inflexible. I'm lucky in being able to call in quickly, and a revalidation takes just a minute to do, but it is worth bearing in mind if you are on a S-Bahn in Munich with the clock ticking on you.....