Beitzah 15b - lashem and lachem

The gemara brings a well-known machlokes between R' Eliezer and R' Yehoshua. R' Eliezer says that Yom Tov should be either 'כולו לה or כולו לכם, entirely dedicated to Hashem, Torah and avodah, or entirely dedicated to yourself, simchah and enjoyment. R' Yehoshua argues and says חצי לה' וחצי לכם, half of each.

I am wondering, is there any overlap between their positions? What I mean is, does R' Eliezer also allow חצי לה' וצחי לכם? His words would then mean, there is no requirement to do both, you can do all for Hashem, all for yourself, or anything in between. R' Yehoshua would be arguing to say that there is a requirement to make sure to do both.

I hadn't heard of anyone explaining it that way, but really the alternative seems very strange: that R' Eliezer would require specifically either all this or all that. What would the idea be for such a thing?
Also, in the gemara there, R' Eliezer is teaching Torah on Yom Tov, and criticizes the students who leave early. But for those who stay till the end, he has praise, and then tells them to go enjoy their Yom Tov meals נחמיה ח) לכו אכלו משמנים ושתו ממתקים).
Why would he do that if they are forbidden to have a festive meal after they spent part of the day learning?

Another point is that the gemara continues, "everyone agrees that on Shavuos one also needs לכם." In other words, R' Eliezer "also" needs לכם on Shavuos. Sounds like there he requires both, apparently based on a סברא. But his original din is based on a clash of two pesukim; how can it be nullified by a סברא?
It's much easier if his original din still stands, only the severa whittles it down: unlike other yomim tovim, you have an extra requirement on Shavuos.

As for my original question of R' Eliezer talking to his students, see the Maharsha. He does not seem to be learning like my suggestion.

Also, I'm (somewhat) bothered by the original language of the gemara:
אין לו לאדם ביום טוב אלא: או אוכל ושותה או יושב ושונה
"A person has nothing on Yom Tov except, either eat and drink, or sit and learn." The starting phrase, "a person has nothing on Yom Tov" is obscure to me, but sounds like a restriction rather than expanding on R' Yehoshua's position.