My child's health and well being and comfort have no price. I wouldn't dare tell my child, "suck it up, you have to walk" just bc she is a certain age. This same mentality is why my mother (who is only 60 yrs old) is giving me a hard time about renting her a wheelchair or the motorized thing at WDW. The attitude of others who think and push their view that if you are a certain age, you should walk, make others embarrassed or afraid to admit when they actually do need assistance. My mom's argument is "I'm 60, not 80." or "I'm not an invalid." Truth is, she is a woman who can not walk quickly or for long periods of time without her back starting to hurt. So why should she suffer in Disney to avoid looks or because she's too proud? It's ridiculous. My point is: It shouldn't matter what age you are--- you shouldn't be forced to walk nor should it be assumed you can keep up. 7 yr olds can tire easily and need to be pushed just as much as a 4 yr old. If you are one of those people who is in the best of health and can walk Disney from sunup to sun down and not feel at all worn- then good for you. Consider yourself blessed but don't assume everyone is like that.
So why don't you just walk more slowly and do shorter days? It would probably make her feel better to be able to walk on her own.