Specific answers to your questions...
What would such an action as ignoring an appeal accomplish?
Practically speaking, it could save you a bunch of money if you know and/or accept the imminent outcome and/or can't afford to proceed. Other than that, not much.
What are some possible responses for the appellant?
No response from the appellant would be needed. Once appellant has filed a Notice of Appeal and put the process in motion, the participation of appellee (and indeed of appellant) is based on filing necessary motions and/or documents by their proper deadlines.
Could the appellant notice the court that the appellant has won by default?
"Noticing" the Court of anything would be unnecessary. The Appeals Judge (and/or, often, the Clerk of the Appeals Court) would be aware of the failure of a party to file a brief or appendix, for example, and those instances would provide options moving forward.
Do the appeal courts ignore the whole issue?
Do the appeal courts rule on the paper (single brief) before them anyway?
Yes and no. In terms of briefs, yes, they use the one brief that has been properly submitted. No, they don't use it exclusively. They also have a complete record, including transcripts, from the trial court and they may or may not hear one or both sides' oral arguments. Additionally, in any given appeals proceeding, there could be one or more of the following filed by one or more of the parties: brief, reply brief, appendix, motion, petition, notice, certification, application, or other document.
What kind of trick is this anyway?
No idea what you mean here.
Check the Rules...
Most often, all you need to know is contained in the particular jurisdiction's Rules of Appellate Procedure and the Court's local rules. The processes appeals litigants must adhere to, the ways in which they may do so, and the consequences for failing to do so can, of course,+
`a vary by jurisdiction. That said, here is a representation of the types of Rules which appear in American appellate courts:
If an appellant fails to file a brief within the time provided by this rule, or within an extended time, an appellee may move to dismiss the appeal. An appellee who fails to file a brief will not be heard at oral argument unless the court grants permission.
*Rule 31(c), Federal Rules of Appellate Procedure
If the appellant fails to file an initial brief, the clerk is authorized to dismiss the case.
Rule 31, Federal Circuit Rules
In Maryland, state regulations spell out the deadlines with which to file briefs and the reply brief. It also explicitly states what may happen if a party does not comply. MD Rules, Rule 8-502(a), (c).
The DC Court of Appeals (not federal) Rules, at Rule 13, state, regarding involuntary dismissal:
The court, sua sponte or upon motion of the appellee, with or without
notice, may dismiss an appeal for failure to comply with a rule of
this court or where otherwise warranted.