First, something that was not explicitly mentioned in your question is that in both cases you refer just to indexation, so answer will also be given in context of indexation.
Second, your premise is false. Datasets & research do not use just multiples of 10 but also multiples of 5 often. For example, for HICP Eurostat choose year 2015 as base year. Moreover, you will occasionally see other numbers as well.
There is no reason to choose round numbers or multiples of 10 or 5 other than round numbers and multiples of 10 or 5 are aesthetically pleasing.
- Choice of base year in any index does not matter analytically, it only changes visuals. Analysis won't be affected by converting GDP into 2009 or 2010 or 2011 etc dollars. As long as whole analysis uses the same dollars for everything there will be no change.
- Only time when choice of base year in index matters is when you are doing a visualization. For example, if you want to know how productivity in Russia increased since fall of USSR (in 1991), you could choose base year 1991 making productivity index in 1991= 100 which makes it easier to see if productivity declined or increased just by looking at the number itself (although even if you would choose year 1990 a correct comparison would take no more than few extra seconds of your time). Even here it is context depended as the graphs will look almost identical.
Most of the reasons you state are false except the first one:
These numbers look round and are convenient.
This is actually correct reason in most of these cases. For some reason people like numbers that are multiples of 10 or 5, why this happens is question for psychology not economics.
If you used the benchmark as 1991 or 2011, the readers will think you had some reasons to omit 1990 or 2010, you will be questioned about cherry-picking data.
This is false. If choice of index would make actually a difference for an analysis then choosing 2010 would be equally questionable and people could equally accuse you of cherry picking. In any case where choosing different year would actually affect data analysis, the choice of year needs to rigorously justified or your research will be sooner or later questioned.
Multiple-of-10 years do have an advantage in the USA and some other countries because these are the census years, so the macro-level data would be up-to-date and better in demographical.
- Most countries around the world do not even conduct censuses in round years. See here database for dates of all censuses around the world. For example, the Netherlands where I live conducts census every 10 years with years ending with 1 (i.e. 1991, 2001, 2011, 2021), yet the Netherlands does not indexes its statistics with years ending with 1. For example, as you can see here CBS (Statistical office of the Netherlands), indexes CPI with year 2015 (following Eurostat). You will not see virtually any indexes in year ending with 1 in any research produced by Dutch research centers such as Centraal Planbureau or De Nederlandsche Bank, even when research is only done on Dutch level without any international comparison.
- You can still combine all census data with GDP, CPI or whatever you want being indexed or expressed in dollars from any year you choose. For example, if you want to regress real GDP on population, trying to see if larger population leads to higher GDP, and you have census from round years (e.g. 2010, 2020...) you can still choose to express everything in 1997 dollars or 1998 dollars. It literally does not matter. Only thing that does matter is that once you choose some base year you should express all dollar quantities in that base year and not use let's say consumption in 2005 dollars if income is expressed in 2010 dollars.
To really drive this point home, below I extracted from Fred dataset Consumer Price Index for all Urban Consumers. I made 2 graphs. In one of the graph the benchmark year is 2007 in the other one the benchmark year is 2010. I removed title from the graph telling you which index uses which year as a benchmark. Can you see any difference, can you tell which one is indexed to 2007 and which to 2010?