Open Ruy Lopez
Nxe4 is indeed very legitimate - this is the Open Ruy Lopez and was very popular until the beginning of 20th century. There is nothing fundamentally wrong with it, it's just a bit overexplored. Lines are more tactical and sharper than in the closed (Be7) variation, so subsequently they are narrower, longer and thus easier to (over-)prepare.
A famous game in this variation is Smyslov vs Reshevsky, 1945, in which Smyslov didn't leave his preparation until at least 20th move. In comment Rkovach shared this anecdote about the particular subvariation in the game:
Reshevsky sat for 45 minutes (!) on his 16th move. Shortly after receiving Smyslov's immediate reply, Reshevsky asked for time on Smyslov's clock - the response was "2 1/2 minutes" After several confirmations conviced him that it was indeed 2 1/2 minutes Al Horowitz came up, turned the score so he could read it and asked R "Sammy, have you kept up with the Russian journals? - Smyslov published 22 pages on this line and beats it in every variation"
— Rkovach's comment
More recent example was played in this year Tata Steel: Firouzja vs Abdusattorov, 2024, with draw.
So in the end is a matter of taste and sporting concerns. If you are good at tactics and fine with fighting a long and forcing theoretical battle, Nxe4 is perfect for you - just be aware that these long and forcing lines often fizzle out to quick draws and, that if you mess up, you will lose quickly. Otherwise go for Be7, but be prepared to deal with white having lasting pressure.
6. d4 or 6. Re1?
White side prefers to play 6. d4 to 6. Re1 due to 6. Re1 Nc5. Position is similiar to Berlin variation, just a bit improved for black, since his knight is more active. Resulting positions are very drawish and tend to punish ambitious play for both sides. So if white is happy with quick draw he can go for 6. Re1, but 6. d4 is way more promising.
Popularity
As @IanBush commented the Open variation was played pretty much always - even today. However, since beginning of 20th century it was eclipsed first by popularity of closed variations and today by popularity of Berlin.
I made a quick plot based on Caissabase:

Where y-axes is % of all Ruys and open covers ECO codes C80-C83 and closed ECO codes C84-C99.