Live attenuated vs inactivated or "subunit" vaccines
There is a key difference between measles and influenza vaccines. Measles is a live attenuated vaccine (it's a live virus), just like yellow fever virus and these vaccines provide the very best protection, which is widely considered to be life-long. This has been challenged with yellow fever virus, but it is nevertheless highly efficacious
Influenza vaccine is an inactivated vaccine so the presentation to the immune system is impaired.
Most vaccines are not attenuated because it's risky, i.e. if they revert they'll cause an active infection and thereon can be transmitted. Polio is an exception, which is attenuated, and offers superb immunity.
Moreover, when a virus and attenuated vaccine virus enters a cell it undergoes a conformational change when its infecting the host cell (all viruses do this) and antibodies that bind to the virus at this stage of the life cycle will point blank block infection, i.e. they stop the virus entering the cell. If a virus can't enter the cell - it's dead, the technical word for this is "neutralised". An antibody just binding to the virus doesn't necessarily stop the virus infecting the cell and in limited cases it can enhance its ability to infect (antibody dependent enhancement). An inactivated vaccine has a lot of difficulty correctly undergoing a conformational change to "lock onto" the host cell - it's already dead prior entering the body.
A live vaccine generates a massively better immunity, exposing more of the proteins used in its life-cycle to antibodies and a lot more neutralising antibodies. Antibody memory can be life-long.
I'm not mentioning cellular immunity because that has a shelf-life (arguably six months, probably longer).
Genetic diversity
The issues about low genetic diversity of measles are also valid. Measles is a bit like holding the same two cards in a poker hand - every hand, sooner or later it can't win because everyone knows what the cards are.
Influenza's genetic diversity is really large, it's a bit like choosing from two suites of cards in a poker hand and the solution can be any combination. Thats hard to memorise (for the immune system). Specifically, influenza in particular flips/changes is antigenic coat making it difficult for antibody memory between infections to be effective. Influenza's potent trick of 'flipping', measles can not do. Influenza has a segmented genome and can switch its protein coat by "mixing" its two different key surface proteins involved in the infection life-cycle ('mixing theory'). Its technically called "antigenic shift" and measles can't do it because it has a single stranded RNA genome.
See the bibliography for further details especially Oxford's "Human Virology".
Bibliography
"Human Virology" by John Oxford, Kellam and Collier by Oxford Press Fifth Edition 2016
See chapters on Polio, Measles and Influenza
This is text is the most comprehensive virology text at graduate level virology and covers all the material described above. The only downside is there is no sixth edition. Yellow fever virus is poorly covered here and the vaccine is omitted. Yellow fever virus is in the chapter Flavivirues.
Fields Virology: Emerging Viruses 7th Edition 2021
by Peter M. Howley MD, David M. Knipe PhD, Sean Whelan
This is more specialist, but covers SARS-CoV-2 and yellow fever virus, which are not really covered in the "Human Virology" by John Oxford.
For in depth genomics: Principles of Molecular Virology by Alan Cann is a standard text, but "Human Virology" is sufficient ESPECIALLY for influenza. John Oxford was an influenza investigator
Comment polio vaccine
@JenserCube raised the issue of Polio live attenuated vaccine. This is really a separate question but what they are requesting, I'd prefer to avoid talking about because criticism of the polio vaccine got mixed up with conspiracy theory and of course the current health czar for the USA is a vaccine skeptic.
There is a recent trend toward replacing the polio vaccine with a "subunit" vaccine (the old term), basically a reverse genetics based vaccine. A number of publications are proposing this. I'm personally not aware of a current phase 2 trial (the minimum requirement to say there's a series alternative). Thus I would simply say the polio vaccine has been incredibly successful and arguably came close to global eradication (in which case we'd no longer need a vaccine). So my personal view, whilst aware of the counterpoints, is stay the course, hold our nerve and get shut of this infectious disease once and for all.
I am well aware that this is an active research base towards an alternative vaccine, but my personal view is hold the course of global eradication, because once thats done - polio virus is gone the same way smallpox was erradicated.
I honestly can't compromise a long-term, heartfelt viewpoint for a few upvotes or down-votes particularly moving into an era that could turn decisively 'anti-vaccine'. One current vaccine at phase 3 I helped build the molecular foundations of it, so I've a track-record in reverse genetic vaccine design.