Whilst it may help to have that shake I think it's a marginal difference
I recall reading that there are plenty of proteins in the system if you eat enough anyway so it's all there, the shake may help a bit and so over years it may make a bit of a difference but total food is the key, anyway I'd think your approach is fine
As an aside I've also read posts of people who seem to take pride in having no supps which is ok but why not take advantage of a shake if you can, the egg and oats is fine though
On whole eggs why not eat them and then manage calories elsewhere in the diet if concerned? Always seems like a waste of eggs to me, that's all!