Why does the amount of days in an year on average of the Gregorian calendar only have 4 decimal places (365.2425)?

This question was answered in a 1986 paper by the late professor Noel Swerdlow [1].

Briefly, the 1577 proposal for calendar reform was based on a lost book by then-already-dead astronomer Aloysius Lilius (c. 1510 – 1576), in which he proposed the intercalation of 97 days in 400 years. Moreover, Lilius was not the first to propose that intercalation system, for it had already been proposed in 1560 by Petrus Pitatus of Verona. In his work, Pitatus stated that his proposal was consistent with the tropical year of the Alphonsine tables and the mean tropical year of Copernicus' "De revolutionibus" and Reinhold's "Prutenic tables". How could that consistency be achieved? Simply by expressing the fractional part of those values in sexagesimal notation and rounding it to seconds (i.e. 1/3600) of a day:

Alphonsine tables: 365;14,33,9,57

De revolutionibus: 365;14,33,12,54

Prutenic tables..: 365;14,33,9,24

Rounded to 1/3600: 365;14,33

From that:

14 x 60 + 33    873    97
------------ = ---- = ---
    3600       3600   400

Notably, the 1577 proposal to the Pope contained two possible intercalation schemes: the 97/400 and a variable one, in which the decision on whether the first year of each coming century would be leap or regular would be taken on the go as that century approaches. The latter was rejected on the grounds that it was both astronomically uncertain and superfluous to the calendar in any case.

Therefore the adoption of a 97/400 intercalation system was quite reasonable, for the purpose of accuracy, on the basis of the current state of knowledge of Western astronomy, and actually more accurate than the precise value of the mean length of the tropical year in that Western state of knowledge.

In contrast, the 8/33 intercalation system proposed independently by the Syrian ex-patriarch Ignatius N'amat allah who stood for the oriental church in Pope Gregory's calendar commission [2] and by John Dee in England, while closer to the real mean length of the vernal equinoctial year, was not supported by contemporary Western science but by the work of Omar Khayam.

Moreover, replacing the Julian calendar by the Gregorian and not by a radically different one was most logical in view of its foremost purpose, which was not civil but ecclesiastical: the determination of the date of Easter. Since the Church had an elaborate system for that purpose (the Paschal “computus”) which was based on the Julian calendar and lunar tables, it made perfect sense to adopt a calendar that was as close as possible to the Julian. Quoting [1]:

The Gregorian Calendar is not really a new calendar [...] but a number of small “corrections” (aequationes) to the julian calendar and to the Nicaean Ecclesiastical Calendar, constructed on the julian calendar and already in use for over a thousand years at the time of the calendar reform. The object of the reform was (a) to correct errors that had accumulated since the Council of Nicaea and (b) to prevent errors from accumulating again in the future, while altering as little as possible the Nicaean Ecclesiastical Calendar and the method of determining the date of Easter (*)

(*) Note: I object to calling the Ecclesiastical Calendar "Nicaean", as all that Council said was that all churches should celebrate Easter on the same date as the Romans and the Alexandrians, without providing any rules to determine that date.

References

[1] Swerdlow, Noel M. “The length of the year in the original proposal for the Gregorian calendar.” Journal for the History of Astronomy, Vol. 17, no 2 (1986): 109-118. https://legacy.adsabs.harvard.edu/full/1986JHA....17..109S

[2] Bien, Reinhold. “Viète’s Controversy with Clavius Over the Truly Gregorian Calendar.” Archive for History of Exact Sciences Vol 61, no. 1 (2007): 39–66. http://www.jstor.org/stable/41134239.

The other answers have touched some very good points, but there is a mathematical point I would like to make as well: The way the leap year rules are designed is fundamentally incompatible with the decimal expansion. Specifically, rules of the kind "leap/no leap year every X years" don't really care about the number of digits, they are more of a form of continued fractions.

In particular the fact that the decimal expansion has only finitely many digits is both a happy accident in the first place and an unintended consequence of a deliberate choice made afterwards:

1. The error made when setting the length of a year to 365 days is very close to 1/4 of a day, so for the Julian calendar the choice was made to include a leap year every four years, leading to 365.25 days per year on average. If it had been close to 1/3 instead, they would have chosen a leap year every three years, leading to 365.333333333...., an expansion with infinitely many digits. The same for any other factor that has prime divisors other than 2 or 5. In fact you can have a look at the Hebrew calendar, which is based on a complicated 19 year cycle for a real world example of this.

2. When the Julian calendar then was updated to the Gregorian calendar, a deliberate choice was made to keep the rules simple to calculate for a given year. Dropping a leap year every 132 years, would have had almost the same effect as the current rule of dropping one every 100 and adding 1 every 400, with an average day of 365.2424242424...., again with infinitely many digits. However such a rule would have made it incredible tedious to calculate if a given year is a leap year or not, while the current rule only involves calculations that can be made mentally within seconds. But those calculations again are easy because the numbers involved only have prime factors 2 and 5, which again result in a finite number of digits. (Another happy accident here by the way, something like 1 in 300 would have lead to infinitely many digits again.)

The existing answers are good, but I would add one additional detail, building on your concept of "future-proofing":

The exercise of reforming a calendar was intended to make a calendar more accurate, but was not intended to eliminate the need for any theoretical future adjustment.

The Gregorian reform was inspired by the Julian reform, and the Julian reform of the Roman calendar was merely the last of many "consular" modifications of the calendar. Under the Roman calendar system, it was known and accepted that imperfections in the length of a year would accumulate over time and need to be corrected for; the Julian reform was meant to make those corrections necessary much less frequently, but did not pretend that it would make them unnecessary forever.

The Gregorian reform was a further improvement - but if it too was imperfect, and would require someone far in the future to add an intercalary day to a year to bring the calendar back into alignment with the true year...well, that was some future Pope's problem.

The Gregorian Calendar was introduced (to the Catholic World) in 1582, the result of preparation over the preceding five or so years.

However the popularization of decimal fractions would wait another three years until the publication of La Thiende [The Tenth] in 1585 by the Flemish mathematician Simon Stevin. Though not the inventor of a decimal representation of fractions, the publication in 1585 of both La Thiende and La Disme [The Decimal]) popularized them and explained their use.

An important consideration in the preparation and adoption of the Gregorian Calendar was that it be easily understood by those not of great mathematical sophistication. Using a description reliant on an unfamiliar mathematical notation would not have aided the cause.

Note that our entire concept of "number" has changed dramatically over recorded history. A striking observation from Euclid's Elements is how the portion dealing with what we now term Number Theory is discussed entirely in terms of "length of a line segment or arc". That was "number" to Euclid and contemporaries.

The very acceptance (in Europe) of negative numbers dates only to the 13th century (and may comingle with the simultaneous development of double-entry bookkeeping) - so is barely 300 years old at the time of the Gregorian calendar reforms. One must take great care in interpreting historical concept of "number" to not overlay our modern understanding and interpretation.