The tickets are only needed for Rosh Hashanah and Yom Kippur, and yes, they do serve a purpose. Two purposes, in fact.
First, there's the matter of seating. If a synagogue has 2,000 seats and 2,100 people show up, 100 people are going to be stuck without a place to sit. People who attend synagogue year-round are often used to completing their individual morning blessings at home, then coming to synagogue about half-hour into the prayers so they can do just the community-based prayers in community. People who come once a year may show up right on time. Those who attend and support the synagogue all year long may rightfully feel put out if these once-a-year Jews take their seats. On the other hand, once-a-year Jews may also feel rightfully put out that the one time they do want to be there, the seats are all taken by folks who can be there any old time they want (completely forgetting that they, too, could be there every week if they wanted).
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Second is the matter of simple economics. Every synagogue has bills to pay. There's light/electricity, water, heating gas, cooking gas, cleaning staff, building maintenance, security (many synagogues need this, especially in communities with a history of anti-Semitism), library acquisitions, prayer books /kippot/tallitot for visitors who need to borrow them, wedding and b'nai mitzvah funds for those whose families can't afford to celebrate their simchot well, paying the clergy and the office staff...
Full members of the synagogue won't need to buy tickets because tickets come with their membership dues. My synagogue keeps its dues on a sliding scale, with an individual's dues starting at $300 per year (though if one can't pay dues, they never, ever deny membership benefits), or $500 for a couple or family with less than 3 children under 18. That includes High Holy Day tickets for the full family, plus all the classes offered by the synagogue, the right to marry and celebrate other life cycle events there, and an automatic minyan when one needs to sit shiva -- someone dies, and people will flock to surround them with support. You get a LOT more out of a full membership than just the holiday tickets, and I highly encourage everyone to take full advantage of what your synagogue has to offer.
Full, year-round members make up about 80% of the membership of an average Orthodox synagogue, from what I recall, but only about 10-15% of the membership of an average Reform synagogue. Nevertheless, whether everyone or no one comes to the synagogue year-round, the bills do have to be paid year-round, and that's why they ask once-a-year Jews to buy tickets. Those tickets take care of the 85% or so of the funds that are needed year-round.
Some synagogues do offer their High Holy Day seats for free. There's a Conservative congregation in New York that rents out a truly enormous public space for the High Holy Days, and invites the entire world to attend. Some university Hillel Houses will also offer their services free to Jewish university students, and at a very low price for others (around $30 to $50, last time I checked), because they're subsidized by outside contributions. Check a local uni.