But I can't imagine that in a society as competitive as ancient Rome a politician could only rely on the funds of his family and friends.

You'd be wrong to not imagine that, if anything society encouraged such behavior from the senatorial class. The debts accrued by the senatorial class were not because these men needed the money, or rather they were not because they didn't necessarily have it. In some cases they genuinely didn't have the money, particularly for large purchases--land purchase in particular was often done more or less by credit among the senatorial class and the equites, who traded land with each other almost as a sort of hobby. But the cost of political office was not limited to the costs of public largesse and entertainments such as those that you mention--mind you, you're mixing up two incidents, as Caesar's public works were done during his aedileship, not his bid for the consulship or as pontifex maximus. But in any case it is certain that running for senatorial magistracy or really any elected office was an expensive undertaking at Rome, by default. There was the need to allot payments to divisores for distribution to the citizen electorate, and even in the cases like Cicero's election to the consulship that did not involve (on Cicero's part at least) bribery were supremely expensive. Bribery, or ambitus, was a common problem, particularly since the limits of legitimate payments were always something of a legal grey area (and most distributions were done through other parties: divisores, clients, and friends, who themselves were being politically attached to the candidate through the distribution of largitio and ambitus), but to ensure the electoral consideration of collegia, vici, and other citizen associations--always more important than going straight to the individuals in Roman politics, given the way the comitia worked--there was always a certain amount of token payment involved, and the need to, for example, constantly meet with the magistri vicorum and collegiorum that is mentioned in the Commentariolum Petitionis and to continually maintain a major public presence was going to run up a major bill no matter what. Nobody got out of election to public office, particularly to the praetorship and consulship, without taking on massive debts.

Moreover, the Roman senatorial class even when idle traded loans with each other constantly. In a society as politically competitive as that of the late Republic debts served to cement political ties. These debts might be figurative debts of gratitude or real debts such as those that Cicero took on nearly constantly as he shuffled around his property or allied himself with various public figures. Often these debts were, for all intents and purposes, ignored--especially since often both sides were indebted to each other!--but one way or another they represented a major form of the sort of networks of mutual obligations that ruled the political world of the Roman Republic, and managing these debts was a part of the political process, even for those who were not "dirty."

A further note: novi homines were decidedly not plebeians entering into a patrician world. By the late Republic the patrician order had long lost its political monopoly, following the resolution of the Conflict of the Orders in the mid-fourth century, which established the parity of the patrician and plebeian nobility, and by Caesar's lifetime the order had more or less ceased to exist altogether--there existed by then only fourteen families with surviving patrician lines. Of the nobiles, those individuals with consular ancestors, at least half were plebeian, by law: the Lex Licinia had long established the legal requirement for at least one plebeian consul, and in some years both consuls were plebeian. The great names of the gentes Sempronia, Licinia, Junia, Porcia, Hortensia and countless others were all plebeian, and therefore Pompey, Crassus, Cassius, Brutus, Cato, Hortensius, and both the Gracchi are all plebeian names--and none of them are novi homines! Indeed, it's difficult to find patricians at all in the late Republic. There are a few--Sulla, Caesar, Catiline--but they're rather uncommon by comparison to the absolutely enormous number of plebeian names we see filling the ranks of the senate--while the nobiles contained a decently large number of patricians the senatorial class as a whole was almost entirely plebeian

The lavish displays you speak of are a reference to the position of Curule Aedile. The Curule Aediles were responsible, among other things, for the Roman games. While there was a budget for the Roman games, Aediles were expected to also provide extra money to make the games as lavish as possible, or they would be looked at in a bad light. This was not the image you wanted to have if you wanted to run for higher positions. Also, the more money you spent also tended to put you in a very good light to the voters, so most who could did indeed spend the extra money.

The debt you speak of for Julius Caesar needed Crassus to bail him out, however, was not from debts from the games. This was from Julius Caesars debts accrued from bribing the other electors in his bid to become Pontifex Maximus. This was a gamble to gain power by Julius Caesar that ended up paying off when he won the position.

I'm on mobile, so I can't provide sources at the moment but I hope this helps!