Ten years ago First were running Great Western, Thameslink (as First Capital Connect), Scotrail, and TPE, and reckoned they were capable of running anything else they might win. E.g. they tried for South Western and East Coast (and others, I think), and would have won West Coast if the DfT hadn't messed things up. So unless they are now more cautious than they used to be I doubt they hold back from bidding for West Coast if by then they'd also managed to get hold of East Anglia and South Western.
I'd say those multiple franchises were in a quieter era and were less of a management challenge.
They already have plenty to do with GWR and TPE and all the issues of new stock and route upgrades, especially on GWR.
East Anglia is expected to be a tough ask too, with a lot of change in prospect.
Managements have to assess the corporate risk - it's a lot of eggs in one basket.
First Group's finances are regularly under pressure.
DfT have to make a similar assessment of "deliverability".
They are known to want to spread franchise ownership wider to reduce the risk of failure.
I don't doubt that First would want West Coast ahead of EA and SW, but they put the other bids in!
Of course they might not win either, in which case it will be all out for WC.